Linear Operators in Spaces with an Indefinite Metric (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

In this connection, criteria for the discontinuity of plus-operator acting in Krein spaces are of interest. We recall that the symbols P, (i = 1, 2) denote the canonical projectors (see 1.§3.1). Theorem 4.6: In order that a plus-operator V acting from a J,-space .°, into a J2-space W2 should be bounded (should admit closure) it is necessary and sufficient that the operator Pz V should be bounded (should admit closure).

It follows from Theorem 4.3 that the inequality 11 p2+ I'xllz+µlixlli>IIPZ VxII2 holds for arbitrary x E CA,,.

(4.1)

§4 Plus-operators

119

Let Pz V be a bounded operator. Then II Vx112' II Pz Vx112+ 11 Pi Vx112 < (II Pz VII +, II pz VI12+µ) II x111.

and therefore the plus-operator V is also bounded. Conversely, if V is a bounded operator, then it is obvious that Pz V is also a bounded operator. Now suppose that the plus-operator V admits closures, i.e., if follows from y and x -i B that y = B. We verify that Pz V also admits closure. Let Vx,, Pz Vx - y+ and x - B. If the sequences (P2' V) and are fundamental, then by (4.1) is fundamental. Since the operator V admits closure, 0, and therefore Pz Vx 9, i.e., y+ = 0, and the operator Pz V admits Vx,, closure.

Conversely, suppose Pz V admits closure. Then V has the same property. 0, then Pz Vx,, - Pz y, and the fact that PP V can be closed implies that P2 +y - 9; and then it follows from (4.1) that Pi- Vx 0, i.e., y = 9.

For, if Vx - y and x,,

Corollary 4.7: Under the conditions of Theorem 4.6 let .'2 = H. be a Pontryagin space. Then the following assertions are equivalent:

a) V is a continuous operator; b) the operator V admits closure. The eigenvalence of assertions a) and b) follows directly from Theorem 4.6 when we take into account that a finite-dimensional operator (in our case, Pz V) admits closure if and only if it is bounded.

Corollary 4.8: Under the conditions of Theorem 4.6 let 1 = H. and Y2 = IIX be Pontryagin spaces, let cl v = .,Y1 and ,? + ( 1) fl Ker V = 0. Then V is a continuous operator. By Theorem 4.6 if suffices to verify that Pz V is a bounded operator. Let JejJ7 be a basis in.,Yz . Then the operator Pz V can be expressed in the form of a sum x

Pz Vx= i pi(x)el,

(4.2)

i=1

are linear functionals defined on 1v. Continuity of P2 V is equivalent to the continuity of the whole set of functionals (,pi};. Therefore if, where (pl, Ip2,

under the conditions of the Corollary, the operator Pz V were unbounded, would be unbounded. Suppose for definiteness_that pi is unbounded. As is well-known ([XIII] ) Ker ,pl j 9 v, and since lv = .-W1, so ,Yi. By Lemma 1.9.5 there is at

then at least one of the functionals 01, s02, ....

least one positive K-dimensional subspace 91+ in Ker pl (C -Qv). From (4.2) we

have, on the one hand, dim Pz V2+ < x; but on the other hand, since .y,+ + (1) n Ker V = 0, we have dim Pz VY+ = dim V2+ = dim 2+ = x; so we have a contradiction.

120 2

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

In this paragraph we shall assume, if nothing to the contrary is stated, that

V is a continuous strict plus-operator defined on the whole of a J,-space W .Ye, = i and acting into a J2-space Y'2 = Wz O+ i'z, where, as before, J; = P1 - Pi, and P,± are the ortho-projectors from ; on to JY;± (i = 1, 2).

Definition 4.9: A strict plus-operator V is said to be doubly strict if VC is also a strict plus-operator.

We point out that, in general, V` will not be a plus-operator for every plus-operator V (not even if V is a strict plus-operator) (see Exercise 32 below).

4.10: A strict plus-operator V will be doubly strict if and only if V* is a strict plus operator. It is sufficient to use the Definition 4.9 and formula (1.1). Theorem 4.11: the inequality

If V is a strict plus-operator, then there is a S > 0 such that (4.3)

II Vx112 %611X11,

holds for all x E ,0+ (,W, ). The number

it, (V) (II V112+µ+(V))1/2 + 11 VII

can be taken as S. Let X E .:P + (.Yt', ),

11 x 111 = 1. From the strictness of the plus-operator V

and Corollary 4.5 we conclude that the form [ (V ` V - µ+ (V )I, )x, y], is non-negative in .Y°,, and therefore we obtain from the Cauchy-Bunyakovski inequality for any y E YP, with 11 y I1, = 1 that

I [(V`V-h+(V)I,)x,Y]i1z
II(V`V-µ+(V)Il)xll < (II V112+µ+(V))11 Vx11z (for, if (V` V- µ+ (V)I, )x = 0, this is obvious, and in the opposite case we put

y = J,(V`V-µ+(V)I,)xl 11 (V`V-µ+(V)I,)x11, in the above inequality). On the other hand, II (V`V

-µ+(V)I,)XII' ,> µ+(V)- II

V1111 Vx112

and therefore µ+ (V) - II

VII II Vx112

(11

Vx11z, V112+,U+(V))"211

which is equivalent to the inequality II Vx Ilz

µ+(V) (II V112+µ+(V))12+11 VII

§4 Plus-operators Corollary 4.12:

121

Let V be a strict plus-operator, with J v = 01. Then II Pz Vx 112 > (b1,211 x l11

for any X E .JP+ (.,Yj ).

It suffices to note that, because the vector Vx is non-negative, we have II vx112=IIP2+Vxll2+IIPz vx112<211Pz vx112,

and we then use the inequality (4.3). From Corollary 4.5 and Theorem 4.11 we obtain directly Corollary 4.13:

A continuous strict plus-operator V maps every non-negative

(positive, uniformly positive) lineal ? (C.J"i) homeomorphically on to a .

non-negative (positive, uniformly positive) lineal V-W(C.Y(2).

One of the trivial consequences of Theorem 4.11 is the fact that Ker V is a negative subspace. It turns out that a more exact assertion is also true: 4.14 If V is a strict plus-operator and 9 v = Yep, then the subspace Ker V is uniformly negative.

Since the form [ (V c V- µ+ (V )I, )x, y] is non-negative, we obtain from the Cauchy-Bunyakovski inequality that, when x E Ker V and y = Jlx, [(Vcv-µ+(V)IJ)x,y],12=µ+(V)(x,x)i I

-µ+(V)[x,x]I Il (VcV - t +(V)Ii)ll (x,x)i Hence, if V` V = µ+ (V)I1, then Ker V= (B}, and in the opposite case

- [x, X],

11 VcV_ µ '(

V)111

II X 1I

i

,

which proves our assertion.

Let T be an arbitrary linear operator acting from a Ji-space 'Yj into a J2-space W2 and let Pi -q T C 9r, or, what is equivalent, P1 Jr C 9 T. Then the operator T can be represented in the form of a matrix (4.4)

T= 11 Tii 11;.i= 1,

where

T11=P2TP1 IWi,

T12=P2+ TP1_ I-Wi,

T21 = Pi TPi

T22 = Pi TPi

l

.

,

1 ,-WI

.

We note that in the case when the J,-space .,Yi coincides with the J2-space .W'2, i.e., ., = 1(2, Ji = J2, the matrix (4.4) coincides with the matrix (2.1).

Theorem 4.15: Let V be a strict plus-operator and .c1 v = Y1. Then for all -WE .1I1+ . 1) the deficiency def V2 = dim WZ TPi VP is the same. 1

Let Y E .,tl+ and let K be the angular operator of the subspace Y. By

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

122

formula (4.4) the matrix II Vi; II j j=, corresponds to the operator V. Therefore

VIP= ((V1 1 + V12K)xt +(V21+ V22K)xi I xi E Pi) and

P2 V9= ((V11 + V12K)xi I xi E.Wi ), P2 V!= (V11 + V12K).

i.

(4.5)

By Corollary 4.12 we have II(V11+V12K)xi 112

X1+ +Kxi III

(,6/2IIx; 11 1)

and therefore

Ker(V11+V12K)=(B),

(4.6)

and moreover the norms of the operators (V11 + V12K)-1 (see (3.1)) are uniformly bounded with respect to K E ' (=. i ( i , (4.7) < .2/5. It follows from formula (4.5) that the deficiency of V/ coincides with the V12K)(- 1) II

II (V11 +

co-dimension of the subspace (V11 + V12K).Y1+ in,7t . Let Ko be the angular operator of the subspace Yo E Jt (i'1), and let K be an angular operator such that II K- Ko II < (5/,2) II V12 II . Then, from the from the fact that (Iz + V12(K- Ko)(V11 +

(V11 +

V12Ko)(-1))(V11 + V12K0)-01

and V12Ko)(-1)

II V12(K- Ko)(V11 +

II < 1,

we obtain that the dimensions of the subspaces ((V11 + V12K).4i )1 (1 .W'

and

((V11 + V12Ko)

1)1n

2

coincide, i.e., the deficiencies of the subspaces VMo and V2 coincide. By virtue

of the linear connectivity of the operator ball the validity of the theorem follows.

The theorem just proved justifies the introduction of the following

Definition 4.16: The number 6,(V)=dim . z IP2 VV (9E ../l+ (.'Y, ))

is

called the plus-deficiency of a strict continuous plus-operator V with 1'v=.W1.

We now introduce a criterion for the double strictness of a strict plusoperator. Theorem 4.17:

The following assertions for a strict-operator are equivalent:

a) V is doubly strict; b) the subspace VI'o E -it' (.W2) for at least one go E . //+ (,;V,);

§4 Plus-operators

123

c) VIE ,!/+(J1'2) for any

d) for at least one angular operator

Ko E Yl i the operator (V11 + V12Ko)-' exists and is defined on the whole of .YPz ; e) for any angular operators K E J 1 the operator (V11 + V12K)-' exists and is defined on the whole of W2+.

We notice at once that from (4.5) and (4.6) the implications b) a d) and

c) a e) follow immediately. Further, c) a b) since, by Theorem 4.15, the plus-deficiency S+ ( V) does not depend on the choice of Y E -//+. It is therefore sufficient for us to prove a) d) and c) - a). a)

d). It follows from Proposition 4.10 that V* = I I (V*)ij I I 4 j = I

((V*)ij = Vji, l,) = 1, 2)

is a strict plus-operator whenever V is, and therefore from formula (4.6) written for V* with K = 0, (i.e., 2 = .Y'I) we have Ker Vi, I W2 = (B1

and

, 7vt, =,ivt,.

Comparing these relations with (4.6) we conclude that the operator V,1 is continuously invertible on the whole of .02, i.e., d) holds when KO = 0. It can be verified immediately that in this case (V) II V-1 II2 <1-inf,I1

V21V11xi 112 Il xz

112

and therefore µ+/z (V) V,,11' is a contraction. c)

a). First of all we prove that V' is a plus-operator. If this were not so,

there would be a vector (0 ?6 ) xo E .sA+(,H'2) such that V'xo E .Y'- - (.Y11). Let Q

be the angular operator of any maximal negative subspace containing V`xo.

Then K(= Q*) is the angular operator of a maximal positive subspace Ji-orthogonal

Theorem 1.8.11). to V`xo (see Consequently Vcxo [1] (Pi + K),Y"1+ therefore xo [1] V(Pt + K)it" , i.e., and (.Y"z)) is J2-orthogonal to the maximal positive subspace

V(P; + K).Y°; (see Corollary 4.13)-we have obtained a contradiction; so V` is a plus-operator. We verify that V` is a strict plus-operator. In the opposite case v C ,0 + (.Yt'1), and since Ker V = i V J (see 1.12) and since by 4.14 Ker V is a uniformly negative subspace, it follows from Proposition 1.7.4, Corollary 1.7.17, and Proposition 1.1.25 that y, is a maximal uniformly positive subspace (with an angular operator K' E it' = X + ( z , .W'z) with 2 into K' II < 1). The operator V' acts from ,z =.YPz Wi and has, as is easily verified, the matrix representation i = JYi (

Vc= II(V`)ijlI .j=1, where (V`)ii= V1*1 (i= 1,2), and (V`)ij = - Vji(i j). Since

V`.YYz CJfv,,

so

-V2=K'Vi'1 and

V22 =-K'V2*1.

124

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

Hence, we have

inf{[V`x,V`x],IXEJ°++(Jr2)}>1-IIK 112inf{IIV`xll1 xECO'++( 2)}. [ x, x1 2 2 Ilxll2 To show that the operator V` is strict it suffices to show that the quantity on

the right-hand side is positive. To do this we note that it follows from the strictness of the operator V and from Corollary 4.13 that the angular operator of the subspace V.Wi , which coincides with V21 VT I', has a norm less than 1, and therefore 11 V,*1 ' V2*1 11 is also < 1. Therefore (see Proposition 1.8.7)

11xE.9++(e2)} )inf{II I`xz2+Kxz

inf{II

11211

X2 EX2+ ' KEW }

IIxII

inf{IPA V2iiX21IKx2)I1 1Ix2E W2 , KE Y2 2

inf{II(Vi1 II 2*,K)x2 11, x2E X2+ 1172

1

> o.

' 211 V*-' 112(>i 011 V*-' V21 11j)

Definition 4.18: A continuous (J1, J2)-non-contractive operator V with 1'v = W1 is said to be a (J,, J2)-bi-non-contractive operator if V` is a (J1, J2 )-non-contractive operator.

It follows directly from this definition that a (J1, J2)-bi-non-contractive operator is a doubly strict plus-operator. However, not every doubly strict plus-operator is a (J1, J2)-bi-non-contractive operator; as an example we may take the operator "P;, which is a doubly strict operator in the J,-space .'1. The following theorem describes all (J1, J2)-bi-non-contractive operators. For a continuous (J1, J2)-non-contractive operator V with J'v = W1 to be a (J,, J2)-bi-non-contractive operator it is necessary and

Theorem 4.19:

sufficient that it be a doubly strict plus-operator. Necessity: As already noted, this is trivial. Sufficiency: We consider the space 'Yi _ -Y2 @,W1

(4.9)

and we introduce into it a Hilbert metric and an indefinite J1-metric given respectively by the formulae (<X2, x1),

[(X2,x1), where x;, y; E J; (i = 1, 2).


(4.10) (4.11)

§4 Plus-operators

125

It can be verified directly that the Jr-metric is generated by the operator

Jr = J2 Q+ - J1

(4.12)

and that this operator is a selfadjoint involution, and the canonical decomposition of WP is defined be the formulae .or = .-Wr Q+ .mar ,

r =f Q+ 'WIT .

(4.13)

Moreover, it is easy to see from formula (4.11) that if T is an operator densely defined in .Yl and mapping into .2 then the Jr-orthogonal complement to its graph FT = (< Tx, x) I X E 1T) in Wj coincides with the graph of

the operator T`:

rT1-11 = rT` _ ( I Y E 9r).

(4.14)

It follows from the same formula (4.11) that the graph r v of the operator V is a non-negative subspace. Moreover, it is a maximal non-negative subspace. For if it were not, then by Theorem 1.4.5 there would be a vector (x2, xz) in r which was Ji,-orthogonal to r v, and therefore it would follow from (4.14) that xi = V`x2. But since V is a doubly strict plus-operator, xi = 0. We now

use Proposition 4.14 and obtain xz = 0. Therefore r v E -It' (,Wf ), and so r E -11-(fir) (see (4.14) and Theorem 1.8.11),

i.e., [Y,Y]2- [V`Y, V`YJi 5 0 for all yE.2 Corollary 4.20:

If V is a doubly strict plus-operator with

[Vx,Vx]23µ[x,x]1 forµ>0andallxE

v = ,i and if ,,

then

[V`Y, VCYJ 3µ[Y, Y]2 for all yE.02,

(or (what is the same thing) V ` V - µI1 > 0 implies VV` - µI2 S 0), and therefore

µ+(V`)=µ+(V) and max(0; µ_(V`)) =max(0;µ_(V)). It suffices to note that (1/,µ)V is a (J1, J2)-non-contractive operator and then to use Theorem 4.19. Remark 4.21: We leave the reader to compare Theorems 4.17 and 4.19, and

to compile a set of criteria for a (J1, J2)-non-contractive operator to be a (J,, J2)-bi-non-contractive operator.

3

Here we investigate the interconnection between certain classes of plus-

operators V. As before, we shall assume that V is continuous and that

(v=,1.

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

126

Definition 4.22: A plus-operator V is said to be focusing if there is a constant y > 0 such that [ Vx, Vx]2 > y 11 Vx

11

z

for x E

Definition 4.23: An operator V is said to be uniformly (J,, J2)-expansive if there is a constant S > 0 such that [ Vx, Vx]2 >, [x, x] + S x li ;, i.e. 11

V`V- I, >' 0, and to be uniformly (J,, J2)-bi-expansive if the operator V` has the same property.

Theorem 4.24: alent:

For a plus-operator V the following assertions are equiv-

a) b)

V is collinear with a certain uniformly (J,, J2)-expansive operator; V is a focusing strict plus-operator;

c)

µ+(V)> max[O;µ-(V));

d)

there is a vo > 0 such that V ` V - POI,

0.

a) - b). If V is collinear with a uniformly (Jii, J2)-expansive operator, then V is a strict plus-operator (see Corollary 4.5) and there are numbers a > 0 and 6>0 such that [ Vx, Vx]2 '>a [x, x], + S jj x, j12. Therefore when x E ? + (JY,) we have X112 >

[Vx,

II VxlI2,

VI

i.e., V is a focusing plus-operator. b) c). Let V be a focusing strict plus-operator:

(xE e+(

[VX,Vx]2>, y I t'xlI2 I

i))

If a-( V) < 0, then the implication b) - c) follows from the definition of the strictness of a plus-operator V: µ+ (V) > 0.

Now suppose µ_ (V) > 0. We consider operators

Ve = Sf2 V, where

_ (1 - e)P2 + P2-. For sufficiently small e > 0 and x E taking Definition 4.22 into account, Sf2)

(.Y,) we have,

[ Vex, Vex)2 = [(V- ePi V)x, (V- ePi V)x]2 Vx, Vx]2 + (e2 - 2e) [P2 Vx, P2 Vx]2

(y+e2-2e)II VxIiz '> (y+e2-2c)II VexII2, and therefore Ve is a focusing plus-operator. Since [ Vex, Vex]

[ Vx, Vx]2 we

have µ-(V) <µ-(Ve)
(1 - e)2 [ Vx, Vx]2

e2 - 2e < 0 for sufficiently µ+(VE) < (1 - 02A+ (V) < µ+(V). (since

small

e > 0),

and

therefore

§4 Plus-operators

127

c) - d). Suppose the contrary: µ+ (V) > max (0; µ_ (V)) but nevertheless there is no v > 0 such that V`V- PI, : 0. Let the positive numbers v,, v2, v3 satisfy the following inequalities: µ_ (V) < v1 < P2 < P3 < µ+(V). Then (see Corollary 1.1.36) [ Vx, Vx]2 > v; [x, x] 1 (i = 1, 2, 3). From the premise about V we have that there is a sequence of vectors [xn) C -WI with II xn II = 1 such that [IX,,, VXn]2 - v2[xn, xn]1 = [(V`V- P211)X,,, Xn] I - 0 as n - oo.

Since the operator V*H2 V - v2 J1 is non-negative, we have

(V*J2V-v2J1)xn--'0.

(4.15)

Let (x) be a sub-sequence of the sequence (xn) consisting of semi-definite vectors, for definiteness, let us say of non-negative vectors. Then [(V`V- v211)Xn,Xn71 > [(V`V- v31l)X.,X,]I, and therefore

If

C .?

0.

(4.16)

(V*J2V- vIJ1)x,,-'B.

(4.17)

then similarly we obtain

Comparing the relation (4.15) with the relation (4.16) or (4.17) we obtain

x B-a contradiction. The implication d) > a) follows immediately from the fact that 1/ vo V is a uniformly (Ji, J2)-expansive operator.

C Remark 4.25: In proving the implication c) = d) it was established, essentially, that if V is a strict plus-operator and µ+ (V) > v > max [0; A- (V)J, then there is a S, > 0 such that [ Vx, Vx]2 v [x, x] 1 + S, x 11 ;. Moreover, the converse proposition is true: if [ Vx, Vx] 2 v [x, x+ 1 + x 11 1 for all x E .01 ,

v > 0, then µ+(V) > v > max(0;µ_(V)). For, µ+(v) >, v > µ_(V) by Corollary 1.1.36. We verify that v cannot coincide with µ±(V). Suppose, for example, that µ+ (V) = P. Then, by the definition of the number µ+ (V), there would be a sequence (x) C + ( 1) such that [xn ,xn] 1 = 1 and which i.e., [ Vxn, Vxn]2 - . ( V), [ Vxn, VXn]2 - µ+ (V) [Xn, Xn] 1 -- 0, contradicts the inequality [ VXn, VXn]2 - µ+ ( V) [X,,, xn] 1 , Sv x 1 3 sv [xn, I = 0, > 0. Similarly it can be verified that µ_ (V) # P. Thus for any plus-operator V (including even non-strict ones) 11

11

V, V- v11:0 (max10;µ_(V)) < v <µ+(V)),

(4.18)

and when max10;µ_(V)) ?6 µ+(V)

V`V- vlt

0 (max10;µ_(V)) < v <µ+(V)).

This remark enables us to prove the Proposition.

(4.19)

128

4.26:

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

For a uniformly (J,, J2)-expansive operator V to be a uniformly

(J,, J2)-bi-expansive operator it is necessary and sufficient that V` be a strict plus-operator. The necessity is trivial. The sufficiency follows from the fact that, by

virtue of Remark 4.25, it, (V) > 1 > µ_ (V), and therefore we have, from Corollary 4.20, µ+ ( V`) > 1 > (V`). It remains only to apply Remark 4.25 again. We formulate next a theorem which characterizes the `power' of the set of uniformly (J,, J2)-expansive operators. Theorem 4.27: The set of uniformly (J,, J2)-expansive operators ((J,, J2)bi-expansive operators) is open in the uniform operator topology in the set of

all continuous operators acting from ., into .12, and its closure in this topology coincides with the set of all (J,, J2)-non-contractive operators ((J,, J2)-bi-non-contractive operators). Let V be a uniformly (J,, J2)-expansive operator [Vx, Vx]2 [x, x], + V I I - I V'- V112>0. Then

S11x11i(S>0),andlet V'besuch that 6-211 V'- VII

I

[V'x, V'x]2 = [(V+ V'- V)x, (V+ V'- V)x]2 [Vx,Vx]2-(211 V'- VII 11 VII+11 V'- VII2)11x11 [x,x]I + (s -211 V' - VII II VII + II V' - VII2)I1 x11 2 , i.e., V' is a uniformly (J,, J2)-expansive operator. If, in addition, V were a uniformly (J,, J2)-bi-expansive operator, then by Theorem 4.17 0 E p (V ), and therefore for V' from a sufficiently small neighbourhood of V we have 0 E p (V(, ). It then follows from Theorem 4.17 and Proposition 4.26 that V' is a uniformly (J,, J2)-bi-expansive operator. Since the uniform limit of (J,, J2)-non-contractive operators is a (J,, J2)-

non-contractive operator, the closure of the set of uniformly (J,, J2)expansive operators ((J,, J2)-bi-expansive operators) is embedded in the set of

(J,, J2 )-non-contractive operator (J,, J2)-bi-non-contractive operators). It remains to verify that if V is a (J,, J2)-non-contractive operator ((J,, J2)-binon-contractive operator), then it can be approximated in norm by uniformly (J,, J2)-expansive operators ((J,, J2)-bi-expansive operators). To do this we bring into consideration the operators

IV)=,1+eP; +,1-cP1.

(4.20)

It can be verified immediately that when 0 < e <, 1 the operators IV) are uniformly J,-bi-expansive and when c -, 0 they converge in norm to the unit

operator in,. Since [ VIf')x, VIe'>x]2 Vie')

[Ie')x, If')x] _ [x, x] + e II x 11 i,

are uniformly (J,, J2)-expansive operators ((J,, J2)-bi-expansive

§4 Plus-operators

129

operators) which as c --+ 0 approximate in norm the (J,, J2 )-non-contractive ((J1, J2)-bi-non-contractive) operator V. From Theorems 4.24 and 4.27 follows immediately

The set of focusing strict plus-operators (focusing doubly strict plus-operators) is open in the set of all continuous operators and its closure coincides with set of all strict plus-operators (doubly strict plusCorollary 4.28:

operators). We close this paragraph with the Remark 4.29: Plus-operator and sub-classes of plus-operators acting from the anti-space to . i into the anti-space to k2 will be called (as operators acting from W1 into .W2) minus-operators, and the names of sub-classes will be changed correspondingly (for example, (- Jl, - J2 )-non-contractive operators will be called (J,, J2)-non-expansive). All the propositions given above for plus-operators can be reformulated in a natural way for minusoperators. We leave the reader to do this, and later, in using such propositions, we shall refer back to this Remark 4.29.

4

If V is a continuous plus-operator acting from a Jl-space .i, into a

J2-space 2, with ci v = .1, then the operator V` V acts in 01, and therefore it is proper to ask about the description of its spectrum.

Theorem 4.30: Let V and V` be continuous plus-operators. Then

a(V`V)>0. From (4.18) and Theorem 3.24 it follows that a(V`V) C IR. Let 1, such - a E a(V`V), a > 0. Then there is a sequence C or,, 11 x that

(n-- co).

(4.21)

Without loss of generality we may suppose that the sequences [ [ x,,, x"] 1) [ [ Vx,,, 1)I' converge and have the limits 0, y, i and [ [ V` Vx, V`

and S respectively. From (4.21) we obtain (by mulitplying (V ` V+aI, )x scalarly by J, x,, and taking the limit as n oo) y + a/3 = 0 and (multiplying (V ` V + aI, )x scalarly by J, V` Vx and taking the limit as n -p oo) S + Cry = 0.

The first equality implies -y > 0; for, if R > 0, then y > 0 because V is a plus-operator, and if a < 0, then y > 0 because a > 0 by hypothesis and y = - 0. Now from S + ay = 0 and the fact that V is a plus-operator we conclude that S > 0, i.e., S = -y = 0 and therefore 0 = 0 also. Hence, lim

[(V`V-µ+(V)Il)x,,,x],=y-µ+(V)0=0,

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

130

and therefore (cf. proof of Theorem 3.24) (V c V - µ+ (V )Il )xn -p 0 as n oo. Comparing the last relation with (4.21), we obtain xn - 0-a contradiction. We now suppose that V is a continuous operator acting in a J-space W' with

v=.y'. Theorem 4.31: If V is a uniformly J-expansive operator, then its field of regularity contains the unit circle T = (i 11 i; I = 1). Moreover, V is a uniformly J-bi-expansive operator if and only if T C p(V). Suppose E E 7 and that E is not a point of regular type of the operator V.

Then there is a sequence (x,) C . when n S > 0,

with 11 xn 11 = I such that (V - EI )xn

B

oo. Since V is a uniformly J-expansive operator, we have, for some

((Vxn, Vxn] - (xn, xn]

(xn, (V - SI )xn] + S ((V - U )X., xn]

+ ((V-EI)xn, (V-EI)xn] >s 11 xnlli, and therefore xn - 0 when n oo; a contradiction. Now let V be a uniformly J-bi-expansive operator. From what has just been proved and Theorem 1.16, we conclude that IT C p (V ). Conversely, let V be a uniformly J-expansive operator and let I C p (V ). Since

VV'- I= (V- I)(V` -I)-, (Vcv- 1)(V- I)-'(V` -I), so

V`V- I,>0 Corollary 4.32:

implies W-60.

Let V be a focusing strict plus-operator in a J-space .Y. Then

all E such that max (0, µ _ (V)) < I E < µ+ (V) are points of regular type of the operator V, and they are regular if and only if V is a doubly strict focusing plus-operator.

This assertion follows directly from Theorem 4.24, (4.19), and Theorem 4.31.

Exercises and problems 1

Prove that for every strict plus-operator V there is on c = c( V) > 0 such that for

all plus-operators from an e-neighbourhood (in norm) of the operator V the plus-deficiency is the same (M. Krein and Shmul'yan [2] ). 2

Prove that if

V1, VZ are strict plus-operators and S+ (V,) = S. (V2) (M. Krein and Shmul'yan [2] ).

V, - V2 E v'm, then

§4 Plus-operators

131

3

Verify that if V is a strict plus-operator acting from a J,-space .W'i into a J2-space .W2, then Pz V also is a strict plus-operator. Hint: Use the fact [Pz Vx, Pz Vx]z > [Vx, Vx]z.

4

Prove that if V is a J-non-contractive continuous operator acting in a Krein space .Ye and 1P v =.Ye, then the disc 1 = (X II X II < 1) belongs to the field of regularity of

the operator V,,, D C if and only if V is a J-bi-non-contractive operator (M. Krein and Shmul'yan [21). Hint: Use (4.8) and Theorem 4.17. 5

Prove that the conditions a)-e) in Theorem 4.17 are equivalent to the condition f ) 00ap(V*,). Hint: Use Exercise 4.

6

Let .;V = Y+ O+ .JP- be a J-space, dim Y = dim .-W+, and let V be a semi-unitary

operator mapping .# into .N". Verify that V is a strict, but not a doubly strict, plus-operator. Hint: Use Theorem 4.17. 7

Prove that a continuous (J,, J2)-non-contractive operator V is (J,, Jz)-bi-noncontractive if and only if its graph Fv is a maximal non-negative subspace in the Jf-space ,Yr (4.5) (cf. Rintsner [4] ). Hint: Compare (4.14) with Theorem 1.8.11.

8 A plus-operator V is said to be stable if all operators from a certain neighbourhood of it are plus-operators. Prove that assertions a)-d) of Theorem 4.24 are equivalent to the stability of the plus-operator V (M. Krein and Shmul'yan [5] ). 9

Let V= II V;;II?;=1 be a continuous uniformly (J,, Jz)-expansive operator, with 9 v = .YPj. Prove that the operator will be uniformly (J,, J2)-bi-expansive if and only if 0¢ap(V1*1 ).

Hint. cf. Exercise 5 and Proposition 4.26. 10

Prove that the set of all continuous strict plus-operators acting in a J-space J ' forms a subgroup and that µ+(T, T2) > µ+(TI)A+ (T2) > µ_ (T, T2) (M. Krein and Shmul'yan [5] ).

11

Prove that if T, and T2 are strict plus-operators in a J-space .,Y, then 6+ (Ti T2) = 6+ (T,) + 6+ (T2), where, we recall, 6+ (T) is the plus-deficiency of the

operator T (see Theorem 4.15) (M. Krein and Shmul'yan [2] ). 12

Prove that every strict plus-operator in the space II. is doubly strict (Ginzburg [2)).

Hint: Cf. Exercises 4 and 5, using the fact that II, is finite-dimensional. 13

Let V be a (J,, Jz)-non-contractive operator acting from a Pontryagin space H into a Pontryagin space II, , with x = x', and It v = Ih. Then V is a continuous operator (cf. I. lokhvidov [17]). Hint: Verify that the operator V satisfies the conditions of Corollary 4.8.

14

Give an example showing that the condition x = x' in Exercise 13 is essential (Azizov).

15

Let V be a J-non-contractive operator, let .'+ C .,P+, 20+ = M'+

nY[li,

and

dim S'0 < oo. Then V1'+ = Y'+ implies V!'0 = T°+ (Azizov).

Hint: First verify that f°+ C VP°++, and then use the fact that t'0+

is finite-

dimensional. 16

Let .Yy = .Yr+ (D .%(- be a J-space,

-Y.

being separable infinite-dimensional spaces

132

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric with orthonormalized bases (e, )m. (e; )mm respectively. Verify that the linear operator V defined on the basis as follows

Vei =e+ ,, i=0, ± 1,...i

Ve; =e+i,

i= -1, -2,...i

Veo =B

is a J-bi-non-contractive operator and

Y+=CLin[(e,'+ei )0-

(earn E,u+

V'++2'+,

but nevertheless VYO+ ie 2'0++, where Y0+ = 2+ n 2+11 (cf. Problem 15) (Azizov). 17

Prove that if V is a (Ji, J2)-non-contractive operator with J v = .#, and if [Vxo, Vxo] = [xo, xo] for some xo, then [ Vxo, Vy] = [xo, y] for all y E 9 v, and therefore Vxo E V p and V c Vx0 = xo, i.e., xo E Ker(V ` V - I,) (Azizov).

Hint: Use the fact that the graph r v = ( ( Vx, X) I x E / v) is non-negative in the Jr-metric (4.11) and that the vector ( Vxo, xo) (EI'v) is isotropic in it. 18

If V is a J-bi-contractive operator, then a,(V) fl T = 0 (cf. Corollary 2.17) (E. lokhvidov [1]). Hint: Use the result of Exercise 17 applied to the operator V`.

19

Let V be a J-non-contractive operator acting in a J-space Y with / v = W. Prove that Ker (V - EI) C Ker(V` - 41) when E E T; in particular, if V is a J-bi-noncontractive operator and t E T, then Ker (V - EI) = Ker( V` - EI ), and therefore 4 v- t, = . when E o ap( V) (Azizov). Hint: Use Exercise 17.

20

Let V be a J-bi-non-contractive operator, Yo the neutral subspace, and Vxo = Yo. Prove that the operator V induced by the operator V in the )-space .h° _ YY 1 /2'o is J-bi-non-contractive (Azizov). Hint: Use Exercise 17 and Theorem 1.17.

21

Let V be a J-non-contractive operator with f/ v = . , and let Y C Ax(V) fl Ker (V` V- I) and V2' c .T. Prove that Y [1] Y (V) when kµ ;d I (Azizov).

Hint: As in Exercise 8 on § 1, use induction with respect to the parameter p + q, where p, q are the least non-negative integers for which the equalities

(V- XI)Px=B, (V-µI)Qy=B hold for xE', yEY,,(V). 22

Prove that if for an arbitrary J-non-contractive operator V we have Vx = Xx,

Vy=µy and IXI=IAI=1, kite, then [y,x]=0. Hint: Use the result of Exercise 17. 23

Prove that if V is a W-non-contractive operator ( W-non-expansive operator) and Vxo = kxo, Vx, = kx, + x0 when I X = 1, then x0 [1] Ker( V - XI) (cf. Exercise 11 on §2). Hint: Use the fact that [Vx, Vx] - [x, x] is non-negative (non-positive) on l v.

24

Prove that for a a-non-contractive operator V all the root subspaces 2',,(V) (I X I = 1), with the exception of not more than x of them, are negative eigensubspaces (cf. Corollary 2.28) (Azizov). Hint: Use the results of Exercise 22 and 23.

25 A a-non-contractive operator V in a separable space H. can have no more than a countable set of different eigenvalues on the circle T (Azizov, cf. I. lokhvidov [1] ). Hint: Use the results of Exercises 22 and 24. 26

Let ,Yi (i = 1,2) be Ji-spaces. A continuous operator V.-,W - . Z with s v = 01 is called a B-plus-operator if [x, x], > 0, x ;d 6 - [ Vx, VX] > 0. It is clear that V is a

plus-operator. Prove that in the case when .W' = II, every B-plus-operator is a strict plus-operator, but the converse assertion if false ([XVI] ).

§5 Isometric, semi-unitary, and unitary operators

133

27

Verify that in the first assertion of Exercise 26 the condition .YY1 =1I. is essential. Hint: In a J-space which is not a Pontryagin space consider an orthoprojector on to an improper maximal positive subspace and use Proposition 4.14 and Theorem 1.8.11.

28

It is clear that every uniformly (J,, J2)-expansive operator is a B-plus operator. Construct an example showing that even if YP1 = YP2 = rik the converse of this assertion is false.

29

Suppose that V is a B-plus-operator and V` a plus-operator. Show that V` is also a B-plus-operator. In particular (cf. Exercise 12), when W, =.'Y2 = II,,, for every B-plus-operator V the operator V` is also a B-plus-operator.

Hint: Consider an arbitrary FE .,//+ (.)l"2); prove that the subspace (VF)1' negative, and use Theorem I 1.19 (cf. [XVI] ). 30

is

Prove that for a plus-operator V with V('O + (.Y'1) n iv) n p °wY2) ;e 0 it is always true that 31 v C ?+ (.W'2) (Brodskiy [ 1 ] ).

31

Prove that under the conditions of Theorems 4.6 it follows from

-P+,(, r,) n Ker V = 0, 9 v J Wi and the fact that the operator V (see (4.4)) is bounded and V11Jrf t =,,Y2' that V is bounded (Brodskiy [1], cf. I. Iokhvidov [17]). 32

Verify that in a J-space .# = X+ Q+ Y' with dim .R'+ = co the operator V11

V= 11

0

01 1

1-

I

is a strict plus-operator, where V is a semi-unitary operator with V,. + ;d but V` is not even a plus-operator (I. Iokhvidov [XVII]). , - .02 be a strict minus-operator. Then Ker Pi VP; I .Jt"i = [0), and Let V: when 9 v = Ye, the equality .*p2 VP, _ Ye2 is equivalent to the minus-operator V being doubly strict (cf. Ginzburg [2] ). Hint: In proving the first assertion, use the results of Exercise 3 and the equality (4.3), and in proving the second, use Exercise 7 on §1 and Theorem 4.17. The Remark 4.29 has also to be taken into account. Ye+,

33

34

Let

V. .Y 1 -y .02 be a (J,, J2)-bi-non-expansive operator, let 2';(C J';) be

uniformly positive subspaces, W; be the Gram operators of the subspaces TI-L" and let P; be the J;-orthogonal projectors on to Y1J (i = 1,2). Then P2 VP1 I-V[l" is a (W,, W2)-bi-non-expansive operator (Azizov). Hint: Without loss of generality assume that Y; C . Y, (i = 1, 2). Write Vin matrix form relative to the decompositions .0, _ Y, [+] Y[", 2 = M'2 [+] Y?] and calculate the matrices V*J2V(,< J,), VJ, V* (,< J2).

§5. Isometric, semi-unitary, and unitary operators Definition 5.1: A linear operator U acting from a W,-space lei into a W2-space '2 is said to be

I

1) (W,, W2)-isometric if [Ux, Ux]2 = [x, x], when x E iu; 2) (W,, W2)-semi-unitary if it is (W,, W2)-isometric and 9u=7r,; 3) (W,, W2)-unitary if it is (W,, W2)-semi-unitary and Mu = .)2.

In particular, if W, = I, and W2 = 12, then 5.1 is the definition of (ordinary) isometric, semi-unitary, and unitary operators.

134

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

From Definition 5.1 it can also be seen that an arbitrary operator U mapping

a neutral lineal cu into a neutral lineal 11u can serve as an example of a (WI, W2)-isometric operator. Therefore, in contrast to (Hilbert) isometric operators, (WI, W2)-isometric operators in infinite-dimensional spaces can be

unbounded and can have a non-trivial kernel. It is clear that U is a (WI, W2)-isometric operator (respectively a (J1, J2)-unitary operator) if and

only if it is simultaneously a (WI, W2)-non-contractive operator and a (WI, W2 )-non-expansive operator (respectively a (JI, J2)-bi-non-contractive operator and a (JI, J2)-bi-non-expansive operator), and for such operators the corresponding propositions in §4 hold. 5.2 In order that a linear operator V with an indefinite domain of definition should be collinear with some (WI, W2)-isometric operator U, i.e., V= XU (X * 0), it is necessary and sufficient that V should be a plus-operator and a strict minus-operator (or a minus-operator and a strict plus operator) simultaneously.

The necessity is trivial. Now suppose, for example, that V is a plusoperator and a strict minus-operator. Then (see Theorem 4.3) there are constants a > 0 and 0 > 0 such that [ Vx, Vx]2 > a [x, x] I and - [Vx, Vx]2 > -/3[x, x]I when xE qv, and therefore (a -0)[x, x] I < 0. Since _q v is indefinite (by hypothesis), a = (3 > 0, and U = (1 /f) V is a (WI, W2)-isometric operator. Later we shall use the following simple proposition, which follows immediately from the polarization formula (see Exercise 1 on 1 §1). 5.3 For an operator U to be ( W1, W2)-isometric it is necessary and sufficient that [ Ux, Uy12 = [x, y] I for all w, y E c1u.

Remark 5.4: It follows form Proposition 5.3 that if x, y E 9u, then [ x [1] y] a [ Ux [1] Uy], and therefore a lineal Y(C 9u) is isotropic in CAu if and only if U2 is isotropic in 9?u. The possibility was mentioned above of a (WI, W2)-isometric operator with

a neutral domain of definition being unbounded. It turns out that the last condition is not essential; there are even (JI, J2)-semi-unitary unbounded operators.

Example 5.5: Let e _ O+ 0 .e- be an infinite-dimensional J-space with infinite-dimensional let be a maximal dual pair of semi-definite subspaces of the classes h + respectively (see Exercise 4 on § 1.10), let go = + n 9- with dim Yo = 1, and let 9± _ Yo + 2+, where 2'+ are

definite lineals dense in 9+. Then relative to the scalar product ± [x, y] (x, y E `+) the lineals 2+ are Hilbert spaces (see Definition 1.5.9 and Propositions 1.1.23). Therefore there are isometric operators U+ mapping on to r+. We now define an operator U on elements x = x+ + x-, x` E ./%-, by the formula U(x+ + x-) = U+x+ + U-x-. It is easy to see that .W+

this is a J-semi-unitary operator, and that the image of the subspace V+ is the

§5 Isometric, semi-unitary, and unitary operators

135

lineal '+ (not closed in ,Y). From Proposition 5.2 and Corollary 4.13 we conclude that U is an unbounded operator. In this example a sufficient condition for the operator U to be unbounded turned out to be that U.+ was not closed. It turns out that a condition of this sort is also necessary. Theorem 5.6: Let U be a (J1, J2)-semi-unitary operator. Then the following conditions are equivalent.

a) U is a continuous operator; b) U.W' are subspaces; c) 9?u is a subspace.

a) - b) follows immediately from Proposition 5.2 and Corollary 4.13. b) - c) we verify that UM it are uniformly definite subspaces. To do this it

is sufficient (see Proposition 1.5.6) to prove that the subspace U.W' are complete relative to the intrinsic norm I [ Ux, Ux]z 11,2 (x E . i ). But the latter follows immediately from the facts that .ei are Hilbert spaces with the scalar products ± [x, y] 1, and ± [ Ux, Uy] z in . Wl , and U I . i are isometric

operators. Thus U!Pi are uniformly definite subspaces. It only remains to apply Theorem 1.5.7.

c) - a). Since [ Ux, Uy]z = [x, y] 1 for all x, y E M, and M1 is not degenerate, Ker U= (0). Therefore the operator U-1 exists. It follows from the same relation that the fact that U is a (J1, J2)-semi-unitary operator is equivalent to the inclusion. (5.1) U-' C U` Therefore U-1 admits closure and it is defined on the subspace Mu, i.e., U-1

is a continuous operator. Hence if follows by Banach's theorem that U is continuous. Remark 5.7. Essentially it is proved in the implications a) - b) - c) that U.1± and U.Y( are regular subspaces. Corollary 5.8:

Every (J1, J2)-unitary operator is bounded.

Remark 5.9: If . i are Gi-spaces, i.e. 0 ¢ op(Gi) (i = 1, 2), then the relation (5.1) also holds for (G1, G2)-semi-unitary operators, and therefore if Ru is a subspace, then U-1 and U are bounded operators. In particular, all (G1, G2)unitary operators are bounded, and the condition that Ube a (G1, G2)-unitary operator can be rewritten in the form

U-' = U` (= Gi 1U*G2),

Iu= .01,

u=.(z.

(5.2)

2E Now let ,Yi be Ji-spaces, i = 1, 2, and let U = II Ui; II;; - 1 be the matrix representation of a (J1, J2)-semi-unitary (J1, J2)-bi-non-contractive operator.

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

136

By Remark 4.21 and Theorem 4.17 the operator U11 is continuously invertible on the whole of ez, and U21 U11' (=- r) is the angular operator of the uniformly positive subspace U.Yi , and therefore II IF < 1. We bring into consideration the operator z

U(r) =11 U(r )ii 11 i.i=1 = I l r

(I2 -

r*r) - v2

(I2 -

r* (12+ -

r*r) - 1/2

rr*)- 1/2

(12 - rr*)-1/2

I I

(5.3)

operating in the space .J'2 = M2+ c Mj . Using Formula (5.2) it can be verified immediately that U(F) is a J2-unitary

operator, and it follows from 3.21 after a straightforward verification that U(I') is a uniformly positive operator. By the construction of U(F) we have that V = U(r) -' U is a (J1, J2)-semi-unitary (J,, J2)-bi-non-contractive operator, mapping i into Jez , and therefore (see Remark 5.4) also ,fj into z

So V has the matrix representation V = II Vii II?I=1, where V,,: is unitary, Vzz: I -i is a semi-unitary operator, and V12 = 0,

V21 = 0.

Conversely, let Y+ be an arbitrary maximal uniformly positive subspace in -Y2 and let r be its angular operator. We introduce the J2-unitary operator U(F) in accordance with formula (5.3). Let dim Yi = dim Mz , dim I < dim . z , and V,,: e, M2 be unitary, V22: . I Wz a semi-unitary operator, and V21 = 0, V12 = 0. Then V= Vii 11 i J=1 is simultaneously a semi-unitary operator and a (J1, J2)-semi-unitary operator, and U= U(P) V is a (J,, J2)-semi-unitary (J,, J2)-bi-non-contractive operator, mapping l into the given space T+ C'2. We summarize the above argument:

Theorem 5.10:

A one-to-one correspondence has been established between

all triples of operators (r, VI1, V22), where r: w2 contraction, Vi,:. ; - .02+ is a unitary operator, V22:

z

is a uniform

i - .)"

a semi-

unitary operator, and all (J1, J2)-semi-unitary (J,, J2)-bi-non-contractive operators: (r, VI1, V22)

U= U(r)V,

where U(F) is constructed according to formula (5.3), and V= II Viil1?i=,,

V21 =0, V,2=0;

U- (r; V,,, V22), where r = Uz, UI,', V ,

V22 = Pz U(r) ' UPI I . I .

Corollary 5.11: Under the conditions of Theorems 5.10 the operator U is (J,, J2)-unitary if and only if V22 is a unitary operator. Corollary 5.12: If U= II Uii 1I?i=I is a (J,, J2)-semi-unitary (JI, J2)-bi-noncontractive operator, then U21 E .9 - U,2 E 99.

§5 Isometric, semi-unitary, and unitary operators

137

This follows from the implication

9- 0 rE9.4*r*E.91'.- U21EY.. If U = U(r) V is a J-unitary operator and if r E .91,,, then

Corollary 5.13:

C\T C AM. Since

Iz -(12

rE,St ,

-r*r)-ins

IZ - (Iz - rr*)- 112 E .gym,

and therefore U- V E 9,,. Since V is a unitary operator, it only remains to use Theorem 2.11, taking into account that the spectrum of a unitary operator lies on the unit circle.

Remark 5.14: In §§4 and 5 we have not so far investigated the spectral properties of J-non-contractive and J-isometric operators, since later, in §6, we shall establish, by means of the Cayley-Nayman transformation, a connection

between the classes of operators and the classes of J-dissipative and Jsymmetric operators respectively, and, in so doing, a connection between their spectral properties.

3

In this paragraph we introduce and describe a special class of J-unitary

operators, namely, stable J-unitary operators. This description is obtained as a consequence of a more general result.

Definition 5.15: A J-unitary operator U acting in a J-space W is said to be stable if 11 U" 11

<, c
We notice at once that if follows from Formulae (5.2) and (1.2) that U-"

11 U*" 11

= 11 U" 11,

and therefore for a stable U we also have

U-"H
Definition 5.16: A group W

.

U) is said to be amenable if on the space

B( 41) of bounded complex-valued functions on U there is an invariant mean i.e., a linear continuous function f w having the properties a) f,,,(1) = 1;

b) fy,(,p) 3 0 when p E B(P1) and cp(U) > 0 for arbitrary UE c) fyl(,p(U )) = fl(,,( U)) = f#(w) for any U E ((. It follows at once from properties a) and b) that for real functions 'P E B( X11) the inequalities inf(p(U) I UE R1) < fy,(wp) 5 sup(,p(U) I UE 411

hold.

(5.4)

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

138

Remark 5.17: Examples of amenable groups are (see, e.g., [XII] ):

a) soluble groups and, in particular, commutative groups b) compact groups and, in particular, bounded groups of operators in a finite-dimensional space, etc. Theorem 5.18: An amenable group ?/ = (U) of J-unitary operators is bounded (in norm) if and only if there is a dual pair (Y+, 2-) of maximal uniformly definite subspaces invariant relative to W, i.e., U2+ C Y+ for all UE W.

Let W be a bounded amenable group, II U II < c < oo when U E Qi. We consider a function ,px,y: U - (Ux, Uy) (x, U E , , U E 9!). This function is bounded on W: I ox,y (U) I = I (Ux, Uy) I < c2 II x II II y II, and therefore it goes into the domain of definition of the invariant mean f v,. It is easy to see that the function (x, y) f-vl (SoX,y) _ (x, y) I is a non-negative sesquilinear form on '. From formula (5.4) we obtain 2

II x

11 2

< inf Iwo,(U) I UE I'?/] <, (x, x)1 5 sup((U) I UE Qi)

c2

x11

i.e., (x, y)1 is a scalar product equivalent to the original scalar product in the sense of the norms generated by them. By property c) of the invariant mean we have (Ux, Uy)1= f*,(,py( . U)) = f,,,(' ,) = (x, y) 1, (x, y) 1, i.e. the group Uis a group of unitary operators in the new scalar product (x, y) 1. Since (x, y)1 = (Wx, y), where W is uniformly positive, ( W"I'2UW-112) is a

group of unitary operators (in the original scalar product), and moreover W lc2

G=

UW-1,2

W- lie

comments for any U E W with the bounded operator which is self-adjoint and continuously invertible

JW- 1,2,

(O E p (G)).

Consequently the spectral subspaces 2+, k- of the operator G corresponding to its positive and negative spectrum respectively are invariant relative to UW-1,2,

and therefore the subspaces q± = W 122. are and all U E W. Since 9+ is orthogonal to 0, it is immediately verifiable that 2+ is J-orthogonal to 2-, 91+ O+ and from the continuous invertibility of W- 1/2 we have . = ¶+ [+]2-. Moreover, since 1F are definite subspaces in the G-metric, ' are definite subspaces in the J-metric. From Corollary 1.7.17 it follows that 2+- are the operators W1 "2 invariant relative to

uniformly definite subspaces. Thus the first part of the theorem has been proved. Now suppose that a set Q! _ (U) of J-unitary operators (it need not even be assumed that )?/ is a group) has a general invariant dual pair of maximal

uniformly definite subspaces (Y+, Y-), i.e., U22 ± C Y ± (U E ?!). Then .0 = 91+ [+] 2 . We introduce in W a new scalar product, equivalent to the original one (see Theorem 1.7.19)

(x,y)1 = [x+, y+] - [x-, y-], where x = x+ + x-, y = y+ + y-, x+, y± E

_T±.

§5 Isometric, semi-unitary, and unitary operators

139

Relative to this scalar product every operator U E R% is unitary. Therefore, if ' is bounded. mii x I I - 1 1x1 1 , 5 M II x II, then II U II < M/m, i.e., the set Remark 5.19: In proving Theorem 5.18 it has been established that a bounded amenable group W = (U} of operators U (we did not use the fact that they are J-unitary) is similar to the group of unitary operators W1"2 UW-1i2,

and therefore the spectrum of each of these operators is unitary, i.e., a(U) C T.

Corollary 5.20: A J-unitary operator U has an invariant dual pair of uniformly definite subspaces if and only if it is stable. This follows immediately from Theorem 5.18 taking into account that the

group generated by a J-unitary operator is commutative and therefore (see Remark 5.17) amenable.

Definition 5.21: A continuous operator T acting in a Krein space M is said to

be normally decomposable if its spectrum a(T) can be split into two non-intersecting spectral sets a(T) = al (T) U a2 (T) such that the subspaces PQ,(T).

and P°,(T°, invariant relative to T (see Theorem 2.20), are

uniformly definite.

Definition 5.22: A stable J-unitary operator is said to be strongly stable if all J-unitary operators from a certain neighbourhood of it are stable. The following theorem links the concepts of normal decomposability for J-unitary operators with strong stability. Theorem 5.23: decomposable.

A J-unitary operator stable if and only if it is normally

Suppose a J-unitary operator U is strongly stable. By Theorem 5.18 and Theorem 1.7.19 we may assume without loss of generality that U± and therefore U is simultaneously also a (Hilbert) unitary operator. We verify that a(U I ,o+) fl a(U I -) = 0. We assume the contrary, and let o E a(U I '+) n o r (U I 4e- ), and let (e'', e`"2) be an open arc of the unit circle containing the point to = e`"°: 0 < cp, < coo < IP2 < 2a. Let E, denote the spectral function of the unitary operator U. We define an operator

U".":: U",", I (E":-E") =toII (E",-Ejp,)

,

and

UP, =I

It follows from the construction of U" , that it is simultaneously unitary and J-unitary, that (E,, - E",).W is invariant relative to U", and that, having

140

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

chosen the length of the arc (e 1, a"z) suitably, we can make the operator U,,, 0,

arbitrarily close in norm to the operator U. Since U.W± = .

+-

,

(EIP2 - EwN)9 _ (E1vz - E,,)'J+ G) (EIPz - E10-

and therefore (E,, - E,,,).' is a Krein J-space, where J = J I (Eu,, - E,,) .N'. To complete the proof of this part of the theorem it suffices to show that in any neighbourhood of the J-unitary operator toI we can find unstable J-unitary operators. For example, the operators to U(I' ), where U(I') is defined by the

formula (5.3) and r ;4 0 will be unstable J-unstable J-unitary operators. Indeed, the operator U(F) is uniformly positive, U(IF) ;d 1, and therefore the spectrum of U(F) contains at least one point (0< ) Xo qd 1, and then XoEo E a(to U(F )) and I Xol o I ;e 1. Hence, it follows (see Remark 5.19) that

to U(I') is not a stable operator. By choosing the norm of the operator r sufficiently small, we can ensure that the operators toU(r) are sufficiently close to EoI.

Now let

a

J-unitary operator U be normally decomposable, i.e.,

o(U) = o, Uo2, 01 n0`2 = 0, and the subspaces '+ = P,,,.-,Y and '_ = P,,.( are uniformly positive and uniformly negative respectively. Since Uq'+ = Y+, so (cf. 1.11 with Formula (5.2)) U2+1I = -+11, and therefore [2'+, 2'+1]) is a dual pair (invariant relative to U) of uniformly definite subspaces. By Corollary 5.20 the operator U is stable. Therefore (see Remark (5.19)) its spectrum is unitary, i.e., o(U) C T. Just as in Corollary 3.12 it can be verified that P and

Pa, are J-selfadjoint projectors. Let P, and I'2 be corresponding nonintersecting contours, symmetric relative to the unit-circle, which surround a, and o2. Then (see, e.g., [VI]) for perturbations Vf, sufficiently small in norm, of the operator U the spectrum of the operator V = U + Ve will also be divided into two non-intersecting sets of and oZ surrounding F1 and F2 respectively. Moreover, if V is a J-unitary operator, then Pa; are J-selfadjoint projectors,

sufficiently close in norm (because of the closeness of V and U) to the projectors PQ; (i = 1, 2), and therefore (see Problem 10 on §3) they are, together with the latter, projectors on to uniformly definite subspaces. It follows from Corollary 5.20 that V is a stable operator, and therefore U is a strongly stable operator.

In conclusion we introduce one more class of operators closely connected with J-isometric operators. 4

Definition 5.24: Let . 'i be Ji-spaces, i,= 1, 2. A bounded operator U: W, W2 with 1'u = Y1 is said to be partially (J1, J2 )-isometric if Ker U is a regular subspace, and U I (Ker U) [11 is a (J1, J2)-isometric operator.

It follows directly from Definition 5.24 that continuous (J1, J2)-semiunitary operators, and, in particular, (J1, J2 )-unitary operators, are partially (J,, J2)-isometric.

§5 Isometric, semi-unitary, and unitary operators

141

5.25: If U is a partially (J1, J2) -isometric operator, then the operator U` will

also be partially (J1, J2)-isometric. Since (Ker U) [11 is a regular subspace whenever Ker U is, and since the operator U I (Ker U) [11 is continuous, so by Remark 5.7 WU = ?UI tier U1111 is a regular subspace, and therefore Ker U` (= W,, [l1) is also a regular subspace. Since U I (Ker U)111 is (J1, J2)-isometric, it follows that U` I a,, = (U I (Ker U)[-L])-1 is a (J1, J2)-isometric operator. It remains only to use Definition 5.24.

Corollary 5.26: The operator U` which is (J1, J2)-conjugate to a (J1, J2)-semi-unitary operator U is a partially (J2, J1)-isometric operator. Corollary 5.27: Let U be a partially (Jt, J2)-isometric operator. Then U` U is a J1-orthogonal projector on to (Ker U)111.

Exercises and problems 1

Let V be a (J1, J2)-semi-unitary operator (not necessarily bounded); then V ' is a continuous operator (cf, [III] ). Hint: Since 9 v = Y1, the operator V` is bounded (cf. [XXII] ). It remains to use the inclusion (5.1).

.Yt'2, then U is a

2

Prove that if U is a (J1, J2)-semi-unitary operator with (J1, J2)-unitary operator. Hint: Use the result of Exercise 1.

3

Prove that if V is a (J2, J2)-isometric operator, and 9v is closed, and _lv is a non-degenerate subspace, then V is a continuous operator (cf. I. Iokhvidov [5] ).

4

Give examples of (J1, J2)-isometric, unbounded, densely defined operators U: .wt'1-.7'2 for which U-' = U`. Hint: Use the device set out in Example 3.36.

5

Prove that an arbitrary (J1, J2)-unitary operator U can be factorized into a product U = U1 U2, where U1 is a J2-unitary operator with U1 = 12, and U2 is a (J1, J2)-unitary operator iwth U2,Yt'1 =,02:-

Hint: Put U1 = U(r) J2 and U2 = J2 V, where U(P) and V are the same as in Theorem 5.10. 6

Prove that if U is a (J1, J2)-unitary operator, then I U _ (U*U)'/2 is a J-unitary operator. Derive, based on this, the polar expansion U= V U1 of a (J1, J2)unitary operator and prove that V is a (Jr, J2 )-unitary operator and V.W; = "Y2' Hint: Use Theorem 5.10.

7

Prove that a bounded group of 7r-unitary operators in a finite-dimensional space has an invariant dual pair of definite subspaces (Azizov, Shul'man). Hint: Use Theorem 5.18 taking Remark 5.17b) into account.

8

Give an example of a J-unitary operator with a single invariant dual pair (.W, .Yt'-) which is not strongly stable.

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

142

Hint: Take as U, for example, the operator U such that U I+ = I+ and U 1.0- is the shift operator along the orthonormalized basis (e; )m in .Y-. 9

Let U U; 11 ?;., be the matrix representation of a J-unitary operator relative to the canonical decomposition .P' = W+ O+ -W-, and suppose there are open sets 1, and S12 such that a

'=

U (a(U,, + U12K) I KE Jt' ) C 0,, 52, n 512 = 0,

92=U (a(U22 + QU,2) I Q E 3

) C 02,

'P(U) C 52, U 512.

Prove that if p (U) fl (E: I E l = 1) * 0, then U is a strongly stable J-unitary operator (Langer [ 11 ] ). 10

Let NI = ( U) be a group of J-unitary operators acting in a J-space W, let Y be its invariant subspace, and let 2i0 = 2' n Y(1] . Then U2'0 = I° (UE Y!). Hint: Use Remark 5.4.

11

Let V be a continuous J-semi-unitary operator, let 2'° be a neutral subspace, and V91° = 2'°. Prove that 2V is also invariant relative to V and that the operator V 2' /2'° (see Corollary 1.5.8) is J-semi-unitary. induced by V in the f space Hint: Use Proposition 1.11.

12

Under the conditions of Exercise 11, let V be a J-unitary operator. Prove that then V is a J-unitary operator (cf. [XV] ). Hint: Use Exercise 20 on §4 and Exercise 11.

13

Prove that if V is a continuous J-semi-unitary operator, then the uniform definiteness of a subspace 2' is equivalent to the uniform definiteness of the subspace VIP.

Hint: Use the definition of a J-isometric operator and the continuity of the operators V and V-' (see Exercise 1). 14

The spectrum of a J-unitary operator U is symmetric about the unit circle, i.e.,

(XEa(U)) a (k-'Ea(U)).

Hint: Use the relation (5.2) and Theorem 1.16). 15

Let U be a J-unitary operator in Y, and let a, be its bounded spectral set, a, I > 1. Prove that then a2 = (ai) ' is also a spectral set for U, that the projector Pa,uo, is J-self-adjoint, and therefore d'=PQ,ua, is a projectionally complete space. Moreover, if of (I ai I > 1) and a2'= (a(`)-' is a second pair of such spectral sets for U, and a, n a' , = 0, and 2' = Po; u0J, then 2' [1] Y'. Hint: Use Exercise 14. For the rest the proof is entirely similar to that of Corollary 3.12.

16

§6 1

V be a J-isometric operator with k, µ E o ( V) and 1\µ * 1. (cf, [XIV]). 2',,(V) [1] Hint: cf. Exercises 21 and 22 on §4. Let

Then

The Cayley-Neyman transformation We shall find it convenient to introduce this transformation not merely for

linear operators but also in a more general setting-for linear relations (see Definition 1.2) be a given Hilbert space. We consider the set 2' of all linear relations Let T, i.e., of all possible lineals of the space ' x _V1. Thus T = (<x, y)) is the

§6 The Cayley-Neyman transformation

143

linear set of ordered pairs (x, y) (x, y E .Ye), and linear operations in ,Ye x Yt' (and therefore also in T) are defined in the natural way (component-wise). We recall (see § 1.1) that a linear relation T is the graph PA of some linear operator M. If and only if it follows from (w, y) E T and x = B that y = 0; so A: that T = rA = ((x, Ax) ).1 E VA (CAA C Jr'). Such linear relations T are called single-valued linear relations. The sets

VT= (xE.YP there is an yE.J'P such that (x, y) E T) 1 T = (y E ,-,Y I there is an x E .) such that (x, y) E T) J

(6.1)

'

are called respectively the domain of definition and the range of values of the linear relation T. For all T E Y we also introduce the following definitions and notation: Ker T= Ix E.WI (x,0>ET),

Ind T= (yE.Y°j(0,y)ET), (6.2)

-T=(-1)T, XT=((x,Xy)I (x, y)ET)(XEC), T-t=((Y,x)I<x,Y)ET).

(6.3)

(6.4)

Ker T and Ind T are called respectively (see Definition 1.1) the kernel and the indefiniteness of the linar relation T. It is clear that X T, T -' E Y. We also introduce the identical linear relation I= ((x, x) ) XE.,y and the sum of two linear relations T1, T2 E ?:

T1 +T2=(<x,yi+Y2) I<x,Yk)ETk, k= 1,21.

(6.5)

It is clear that Tt + T2 E Y and 9T,+ T2 = cI T, n cT,. In particular, we consider the linear relation

Tx=T- XI

(TE22,XEC)

(6.6)

and by means of it we introduce for a linear relation T its resolvent set

p(T)_ (XEC

I

Ker Tx = (0), 3rr_

),

(6.7)

the spectrum

u(T) = C\p(T)

(6.8)

and, in particular, the point spectrum ap(T) = (XECIKer T>,;e (0)).

(6.9)

Points X E p(T) are called regular points, but 9? E ap(T) are eigenvalues of the linear relation T. For eigenvalues X the lineal Ker Ta = (x E Y' I < x, Xx) E T) is called the eigenlineal, and its non-zero vectors are called eigenvectors of the linear relation T corresponding to the eigenvalues X)'.

We can introduce a further classification of points X(E C) of the point spectrum op (T) of an arbitrary linear relation T by dividing these points into '

It will be convenient later, when Ind T ;d 101, also to suppose that, by definition, oo E oo(T)

(Ca(T)).

144

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

two sub-classes ap, (T) and u,2(T) (cf. Theorem 1.16)

up,,(T)= (XEap(T)I.9?r, ;6 W),

(6.10)

(6.11) ap,2(T)_(XEap(T)I_JiT The continuous spectrum ac(T) and the residual spectrum ar(T) of a linear relation T are defined by the formulae

ac(T) = (X E a(T)\ap(T) 191r, # 4rA =

),

ar(T)_ (XEa(T)\ap(T)I4T,*W)

(6.12) (6.13)

respectively

From (6.7)-(6.9), and (6.12), (6.13) it now follows that a(T) is the union of four mutually non-intersecting parts:

a(T) = up,,(T) U ap,2(T) U ac(T) U ar(T).

The field of regularity r(T) of a linear relation T is the set

r(T) = IX E C\ap(T) I T, =.;?T,)

(6.14)

It is clear that p(T) C r(T) (since e_,), and that r(T)\p(T) C arw(T). It is easy to convince onself that the definitions introduced of the sets p(T), a (T ), ap,, (T), ap,2 (T ), ac(T) and ar(T) for a linear relation T generalize the

definitions of the corresponding sets for closed linear operators A. When T= I A the two sets of definitions are simply identical.

2

For a linear relation T E Y we now introduce the direct (Kr) and the inverse

Cayley-Neyman transformations defined for a non-real parameter To do this we first introduce the corresponding transformations element-

wise, i.e., for all pairs (x, y), (u, v) E ' x, kt(x, y) _ (y - ('z, y - ('x),

(v-u,(v-('u)

(6.15) (6.16)

O A direct calculation will verify the proposition. 6.1 The .Ye x.

transformations kt and kt ' are mutually inverse bijections of

onto. xM.

Now for any (k;4)

E C and S, T E 2 we put

Kr(T)= (ks(x,y))<x,y)ET, Kt '(T)= (kt '(u,v))Es.

(6.17) (6.18)

6.2 The transformation Kt((' # (") maps ? on to Y bijectively, and Kt carries out the inverse transformation.

§6 The Cayley-Neyman transformation

145

This assertion follows directly from 6.1. 6.3

Let TE 9 and S= Kt(T) ((' ;d ) or, equivalently, T= KF t(S). Then Ker S = Ker Tt, Ker T = Ker (CS - CI),

Ind S = Ker Tr, Ind T = Ker St,

(6.19)

!2T= R,,

(6.21)

'Ws ='RT"

V, = RT"

(6.20)

The formulae (6.19) and (6.20) follow directly from a comparison of the definitions (6.2), (6.3), (6.5) with (6.17). The same applies to (6.21) for the verification of which the definitions (6.1) must be brought in again. We now investigate how the spectrum of a linear relation T(E') and its

components are transformed on transition from the linear relation T to another linear relation S = Kr(T). To do this we introduce, for (' # (", a mapping 4;t: C

C in the extended complex plane t = C U (co 1:

(X - (' )(X - .)-t, X ;4 (', X * O°, 00,

(6.22)

= oo.

1,

The function µ = ct(X) maps

Theorem 6.4:

ap,1(T), ap,2(T), ac (T), ac(T), r(S), and p(T) bijectively on to ap.t(S), ap,2(S), ar(S), a,(S), r(S), and p(S)

respectively.

When X ;4 (', X ;d oo, we have by (6.17), (6.15) and (6.22)

S,,=S-µI _ <X,y)ETl

(y-('X) [(Y - rX,

-

II

(y-XX)) <X,y>ET}

from which it follows that when

)X E C

Ker S, = Ker Ta,

(6.23) ,s = ?1Ta. We note that, taking (6.22) and Proposition 6.3 into account, the relations (6.23) remain valid for all X E C if, for any linear relation T, we put

Ker T. = Ind T by definition. 1 We point that, on this understanding, the Definitions (6.10) and (6.11) make sense also for a = oo E ap(T) (see the Footnote on p. 143), and so the assertions of Theorem 6.4 remain vald in this case. This is also true when X =,r E up(T).

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2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

In accordance with this we shall also suppose (cf. (6.7))' that

X=ooEp(T)* (Yr=de, Ind T= (0]]. All the assertions of the Theorem now follow directly from a comparison of the formulae (6.23) with the definitions (6.7)-(6.14). Corollary 6.5:

The function

(µ -

1)1,

00,

µ

l,µ # 00,

µ=1,

(6.24)

µ = 00

effects for the linear relations S and T = KF ' (S) the mapping inverse to the mapping ct (see Theorem 6.4).

3 We return to the most important case for us when the Hilbert space W generating the set 22 (C ,,Y x W) of linear relations is a W-space, i.e., it is equipped with an indefinite W-metric [x, y] = (Wx, y)(x, y E) (see 1.§6.6).

Definition 6.6: A linear relation T(E 2') is said to be W-non-expansive (respectively, W-non-contractive, W-isometric) in W if for all (x, y) E T we have [y, y] 5 [x, x] (respectively, [y, y] > [x, x], [y, y] = [x, x]). It is clear that Definition 6.6 generalizes the corresponding definitions for linear operators V (see Definitions 4.2 and 5.1). The latter definitions are obtained from Definition 6.6 when T = F v is the graph of the operator V. In the case when 91 is a G-space, i.e. [x, y] = (Gx, y) (0 g up(G) we introduce Definition 6.7. For a TE 2' the linear relation

T'= ((u, v) E

X W [y, u] = [x, v] for all (x, y) E 71

is called its G-adjoint. Remark 6.8: It can be seen from this definition that T`(E2) exists for any linear relation T. However, even in the case when T= GA is the graph of a

linear operator A ('A, WA C W) the linear relation T` is not necessarily a graph (i.e., a single-valued linear relation)-see Exercise 1 below. However, comparison of the Definitions 6.7 and 1.1 shows that, when T= FA and VA = .W', these definitions are equivalent, i.e., T` = FA'

Definition 6.9: A linear relation T is said to be G-symmetric if T C T`, and to be G-selfadjoint if T` = T. We note that the condition T C T` is equivalent We recall that the condition Ind T= LO is equivalent to T being the graph of a linear operator. But then by Banach's theorem, in the case when this operator is closed, the condition VT implies that the operator is bounded, and this gives meaning to the notation, oo E p(T) in this case.

§6 The Cayley-Neyman transformation

147

in an obvious way to the requirement that [y, x] = [x, y] for all (x, y) E T, and in this form we generalize it to an arbitrary W-space, and call such T W-symmetric.

When T= FA with CIA = ,e the Definition 6.9 goes over into the definitions (of the graph) of a G-symmetric and a G-selfadjoint operator A respectively (cf, with the Definitions 3.1 and 3.2 respectively). Returning to the general case of a W-space we introduce

Definition 6.10: A linear relation is said to be W-dissipative if Im [y, x] > 0 for all (x, y) E T. When T = PA, this condition is equivalent to the operator A being W-dissipative (cf. Definition 2.1). We examine how the Cayley-Neyman transformation affects one or other of the properties of a linear relation T given in the formulae (6.6)-(6.9). The identity (6.25) [Y - ("x, Y - N - [Y - x, y - x] = 4 Im (' Im [ y, x] is established by a simple calculation. From it follows immediately the

proposition. 6.11 Let the linear relations S, T(E.') be connected by the mutually inverse transformations (cf. (6.15)-(6.18))

S=Kt(T) =[
(u-v,'u-('v)I [u,v)ES). Then the following assertions are equivalent: a)

T is a W-dissipative (respectively, W-symmetric) linear relation;

b)

S is a W-non-contractive linear relation when Im W-non-expansive when Im relation for any (' f).

> 0 and is

< 0 (respectively, a W-isometric linear

It is sufficient to compare the Definitions 6.6, 6.8 and 6.9 with the formulae (6.25)-(6.26).

Remark 6.12. It

is sometimes useful to consider the so-called Waccumulative linear relations T distinguished by the condition Im [y, x] < 0 for all (x, y) E T, and the corresponding linear operators. It is clear that only the

factor (- 1) makes them different from dissipative linear relations and operators respectively. Proposition 6.11 can easily be reformulated in terms of W-accumulative linear relations; it is clear from (6.25) that in the reformulation the roles of the conditions Im > 0 and Im < 0 are interchanged. We add further that maximal W-accumulative operators, and, in particular,

J-accumulative operators, are defined in the natural way, and the theory of them is entirely analogous to the theory of maximal W-dissipative (and J-dissipative) operators developed in §2.

148

4

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

In this and the following paragraphs, returning to the main purpose of our

book, we shall specialize our examination still further, restricting it to a single-valued linear relations T (E a'), i.e., to graphs of linear operators. Applying the transformation Kr to T we shall see to it that S = Kr(T) is also a

single-valued linear relation, i.e., a graph. To do this it is necessary and sufficient, in view of the first of the formulae (6.26), to ensure that (' up(T) (see (6.6) and (6.9)). For, if T = rA = ((x, Ax)) XE fA then destruction of the single-valuedness of the linear relation S = Kr(T) is equivalent (see (6.26)) to the simultaneous satisfaction for some x E 2A of the conditions y - ('x = 0 and

y- 'x * 0, where y=Ax, i.e., x ;;6 6 and r E ap(A) (=ap(T)).

Thus, when (f ;e ) Eap(A) we have S=Kr(T)=rv, where V= Vr is a linear operator for which

9v= (Ax- NxEVA;

Rv= (Ax- JXJXEVa,

which can be written shortly in the form

V= I+ ((' - (') (A - CI)- 1.

(6.29)

Moreover, it is clear that 10 ap(V). A simple calculation (cf. (6.26)) shows that here

A=(CV- 'I)(V-I)-'

(6.30)

A = ('I+(('- )(V-I)-'.

(6.31)

or equivalently

It is clear that we can also work in the reverse order, starting with a linear operator V for which 1 ¢ op(V), and for any (;e ) specify the operator A by the formula (6.30) or (6.31). It is also clear that 9A= (VY - AYE V,,

QA= (("VY-f Y)YE C6,

('0 up (A),

and that the formula (6.28) (or (6.29)) is the inverse of the transformation (6.30) (or (6.31)). Traditionally it is precisely in this way that the direct and inverse Cayley-Neyman transformations of operators are defined, and we, allowing a certain freedom, will keep for them the notations V= Kr(A),

A=Kt'(V). The reader will without difficulty reformulate for this particular case Proposition 6.11 and Remark 6.12 which remain valid, of course, on passing from the operator A to its Cayley-Neyman transform, as we shall call, for brevity, the operator V= Kr(A), and reversely. This proposition can be developed further in several directions in the case when it is possible to choose the parameter (f 96 ) so that (' E p(A). In this case

it follows at once from (6.29) that V= Kr(A) is a bounded operator with Vv=.W.

§6 The Cayley-Neyman transformation

149

Theorem 6.13: Let A be a maximal closed J-dissipative operator in a Krein space and let ( ;4 ) (' E p(A). Then when Im > 0 (Im (' < 0) the operator V= Kt(A) is a J-bi-non-contractive (J-bi-non-expansive) operator and

(V - I).

=. M.

Conversely, if V is a J-bi-non-contractive (J-bi-non-expansive) operator and

1 o ap(V), then the operator A = Kr'(V) is when Im (' >O (Im ('
= A = .N' and (- A )` is a J-dissipative operator. Further, by

6.11, V is when Im > 0 (Im < 0) a J-non-contractive (J-non-expansive) operator. From (6.29) taking account of 1.6 and 1.9 we have

V` = I + ( - (')(A` - I)-' = Kt(A`), and the first pair of Theorem 6.13 now follows from Remark 6.12.

Conversely, when 10 up(V) we have, by Exercise 18 and 19 on §4, (V- 1) = 0, and therefore the operator A = V) (see (6.31)) is densely defined and (exactly the same as the operator -A ` = Kt ' ( V`)) it is under the conditions of Theorem 6.13 a J-dissipative operator. Moreover.(see (6.31) and (6.29) (' E p(A), and, by Proposition 2.7, A is a closed maximal J-dissipative operator.

Remark 6.14: It is easy to understand that Theorem 6.13 can be reformulated and proved if the term ` J-dissipative' is replaced everywhere by the term 'J-accumulative' (see Remark 6.12) and the conditions Im (' > 0 and Im (' < 0 change places. If A is a J-selfadjoint operator (A= A') and (f ;6) (' E p (A ), then U= Kr (A) is a J-unitary operator and 1 E ap(U).

Corollary 6.15:

Conversely, if U is a J-unitary operator, 10 ap(U), and (' ;d (', then A = Kr ' (U) is a J-selfadjoint operator. The first assertion follows from the fact that a J-selfadjoint operator A is

a maximal J-dissipative and a maximal J-accumulative operator simultaneoulsy and therefore by Theorem 6.13 and Remark 6.14, for any (' ;d ',

U= Kr(A) is a bounded J-bi-non-expansive and a J-bi-non-contractive operator simultaneously, i.e. (see §5.1) it is a J-unitary operator. The converse argument proceeds entirely analogously. It is not difficult to obtain the result contained in Corollary 6.15 directly without using Theorem 6.13 (cf. Exercise 2 below). 6.16 Under the conditions of Theorem 6.13, the operators corresponding to bounded uniformly J-dissipative operators A (GOA = .i) are when Im > 0 (Im f < 0) bounded uniformly J-bi-expansive (uniformly J-bi-contractive) operators V = Kr(A) and conversely.

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

150

Suppose, for example, that Im follows

from

(6.25)

(when

> 0. Since Im [Ax, x] > yA II x I Iz, it y = Ax) and from (6.28) when

g=Ax-('x, Vg = Ax - 'x that II

Z11g11Z

and similarly for V` (taking account of the fact that for bounded A we have Im [ - A`x, x] = Im [Ax, x] for all x E ,;V), i.e., V is a uniformly J-biexpansive operator. In the converse of this assertion it has to be taken into account that for a uniformly J-bi-expansive operator Vwe have T C p(V) (see Theorem 4.31) and, in particular, 1 E p(V); otherwise the arguments are analogous.

Returning to the general theory of Cayley-Neyman transformations for linear operators A and V (( ), we V= Kr(A) and A = continue first of all the study started earlier (see 6.2) of spectral questions 5

connected with these transformations. With this purpose we shall explain the connection between invariant subspaces of the operators A and V, and also that between the root lineals of these operators.

6.17 LetA be a linear operator in

,

let (' ;e )(" O ap (A ), and let V= Kr (A ).

If the subspace Y C 9A, if A9 C 91 and (' E p (A 12), then ? C 9 v,

VYC9.9 and 1 a(V I _fl. Conversely, if I C "v, V9 C 2', and I ¢ a(V I Y U ap(V), then 2' C 9A, A2' C .' and E p (A 12). Since (see (6.29)) V = I + ( -) (A - (I) ', and by hypothesis (' E p(A 12'),

i.e., (A - ('I)2' _ 2' and 2so 2'C cDv and

V2' C Y. Moreover, V 12 = Kr (A 12'), A 12' and V 2' are closed (and bounded) operators, and from the condition (' E p (A 12') it follows (see Theorem 6.6) that 1 E p (V 12). The converse implication is verified similarly by means of formula (6.31) and Corollary 6.5. Corollary 6.18:

LetA and V be the same as in 6.17, r¢ ap (A) (a 1 ¢ ap(V )).

Then X E ap(A) a (v = (X - )(X - (')-' E ap(V)) and the root lineals /J',(V) and 2',,(A) coincide. The first assertion follows directly from Theorem 6.4. Now let x E 2',(A), i.e., there is p E N such that (A - XI)px = B. Then

2 = Lin (x, Ax, ..., APx) is a finite dimensional invariant subspace of the

operator A: 2'C 9A, AY C 2' and a(A I .') = (X ), i.e.,

(' E p (A 12')

(because (' ;x-1 X). Therefore (see 6.17) 2' C C1 v, V2 C 2' and

(VI 2') = Kr (A 12). By Theorem 6.4 the number P= (X - ')(X- (')-' is the only point of the spectrum of the finite-dimensional operator V1 Y, and so xE Y C 2',(VI 2') C 9',(V), i.e., 2),(A) C . JV). The converse is proved in a similar way.

Remark 6.19:

It is clear that in Corollary 6.18 the first assertion can be

§6 The Cayley-Neyman transformation

151

reformulated equivalently thus: (v E ap(V) } « (X =

3)(v

-1)-'

E op(A)}.

Unfortunately, the requirement that E p(A I 2) (respectively, 1 E p ( V I 2?) imposed in Proposition 6.17 turns out to be extremely stringent. Later, in Chapter 3, we shall see that in particular cases the `preservation' of certain invariant subspaces under the Cayley-Neyman transformation can be ensured under less burdensome conditions. In all the discussions in §6.4 and §6.5 it might have appeared that, in contrast to the arbitrary number }' ( (') on which some requirement such as ¢ op(A) or (' E p(A) was imposed, the number 1, of which it was always required that 1 0 ap ( V), played some special role. But in fact this role Remark 6.20:

can be played by any number c with I e I = 1, if c ¢ ap(V). This leads only to an

insignificant modification of the formulae for the mutually inverse Cayley transformations themselves: (6.32)

-f)(V-eI)-',

(6.33)

and also to corresponding modifications in the formulation of the theorems

and propositions 6.13-6.19, and in particular to the formulae for the transformation of the point spectra in Corollary 6.18 and Remark 6.19: IX E ap(A)) a (v = e(X - f)(X - (') ' E ap(V)), (v E op(V)) a (X = (("v - e)(v - e)-' E op(A)}. 6 In conclusion we consider the Cayley-Neyman transformation V= K-(A) ((' Pd ¢ ap(A)) linking the operators A and V acting in a Hilbert space . = M1 Q+ Mz (a typical situation for a Krein space). We are interested in the case when the operators A and V admit matrix representations (cf. (4.4)) A = I I AtiJI , j=1, V= V;i 11 ?i= 1, generated by the corresponding ortho11

projectors P1 and P2 (Pkr = Yk, k = 1, 2) (for such a representation if necessary that, for example, -1 C 9A and (' E p(A)). In this case the relation

V(A - I) = A - I, which follows from (6.28), can be rewritten in matrix form as V11

A 11 - Ii

V12

A 12

A11-}"I1

A21

A21

A22 - ("I2

A22 - I2 From this we have at once the relations Vz,

V22

I

I

A21

V11 (A 11 - I1) + V12A21 = A 11 -x'11

(6.34)

V11 A 12 + V12 (A22 - rI2) = A 12

(6.35)

Vz, (A u + rI1) + V22A21 = A21 V21 A 1z +, V22 (A22 - rI2) = Azz - (I2.

152

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

In particular, in the case important for us later, when (' E p(A22), it follows from (6.35) that (6.36)

V12 = -(VI I - I, )A,2 (A22 - CI)-'

and if additionally (' E p(A,, ), the elimination of the operator V,2 from the system (6.34), (6.35) gives

Vii [A,, - ('I1 - A12(A22 - 3'I2)-'A21) = Ali - ("It = A12(A22 - 3'I2)-'A2i or

)(Aii - CIi)

(V1t - Ii)[Ii - A12(A22 - ('I2)-'A21(Aii - It)

(6.37)

It is precisely in this form that the relations (6.36) and (6.37) will be used later in Chapter 3 (Theorem 1.13).

§Exercises and problems 1

If T is a single-valued linear mapping into .w', then T` is a single-valued linear mapping into . if and only if Jr = .Y-

2

Derive Corollary 6.15 directly from the formulae (6.29) and (6.31).

3

Prove that Corollary 6.15 can be generalized as follows. Let A be a maximal J-dissipative operator and a maximal J-symmetric operator simultaneously, and let

((' *)1' E p(A). Then V = Kr(A) is, when Im

> 0 (Im

< 0), a bounded

J-bi-non-contractive (J-bi-non-expansive) operator and a J-semi-unitary operator simultaneously and 1 ¢ ap(V). The converse proposition is true, and so are the analogues of both the direct and converse propositions when the word ' Jdissipative' is replaced by 'J-accumulative' and the conditions Im r > 0 (Im 1' < 0) are replaced by Im (' < 0 (Im (' > 0). Hint: cf. Theorem 6.13 and Proposition 6.11. 4

Let V be a W-non-contractive operator in .", let a be its bounded spectral set, and

let P. be the corresponding Riesz projector. Then when I a I > 1 (respectively invariant relative to V is non-negative (nonI a I < 1) the sub space 2 = is positive). In particular, for a W-isometric operator the subspace 2 = P neutral (cf. Theorem 2.21). Hint: To the graph of the restriction V 1 21 (for any f 1) apply Proposition 6.11, Corollary 6.5 and formula (6.31), and then Proposition 6.17 and Theorem 2.21. 5

Let the operator V be the same as in Exercise 4, and let A with I A I > 1 (I A I < 1)

be a certain set of its eigenvalues. Then C Lin(2'a(V))XEA is a non-negative (non-positive) subspace. In the case of a W-isometric V both these subspaces are neutral. Hint: Prove this by analogy with Corollary 2.22, basing the proof on the result of Exercise 4. 6

Let V be a bounded A-non-contractive operator with Cl v = n.. Then a( V) fl (X I I X I > I I consists of not more than x (taking algebraic multiplicity into

account) normal eigenvalues (cf. Corollary 2.23) (Brodskiy [1]). Hint: Use the results of Exercise 5 on §1, Exercise 22 on §4, then Remark 6.19, Proposition 6.17 and Corollary 2.23.

§6 The Cayley-Neyman transformation

153

7

Let n be the set of eigenvalues of a J-isometric operator V and let A fl (A *)-'10. Then C Lin(f,,(V))xEA is a neutral subspace (cf. Corollary 3.14) ([XIV]). Hint: Use the results of Exercise 5, and Exercise 16 on §5.

8

Let U be a ir-unitary operator in II.. Then its non-unitary spectrum consists of not

more than 2x (taking multiplicity into account) normal eigenvalues situated symmetrically relative to the unit circle T (cf. Corollary 3.15) ([XIV]). Hint: Use the results of Exercise 5 on §1, Exercise 22 on §4, Remark 6.20, and Corollaries 6.18 and 3.15. 9

Let U be a a-unitory operator; then a,(U) = 0 (cf. Corollary 3.16). Hint: Use the results of Exercise 8 on §4.

10

If V is a ir-non-contractive operator (1) v = II. ), if k E ap( V) and I k

1, then the

root lineal 'r( V) can be represented in the form ',,(V) =.il"a [+].,ff,, where dim .4", < oo, V,AI",, C,;l"x,, ff,, C Ker( V- XI), and fix is a non-degenerate subspace (in particular, it may happen that .,ff), = (0)). If d, (k), d2(X ), ... , d,, (X) are the orders of the elementary divisors of the operator V 1 SPX, then

Z 2 XEao(V),IXI=1 J=1

[d'(k)I 2

J

+IXj>1 Z

x

(Azizov [8]; cf. [XIV]).

Hint: Use the results of Exercise 5 on §1, Remark 6.20, Corollary 6.18 and Theorem 2.26. 11

Prove that a J-symmetric operator A is J-non-negative if and only if the satisfies the V= Kt(A) J-isometric operator (; ;4 (') Re[I'(J- V)x, x] > 0 for all xE Jv (Azizov, L. I. Sukhocheva).

condition

12

Prove that if V is a J-unitary operator and Re [('(I- V)x, x) > 0 for some f E C and for all x E ., then a(V) C T (Azizov, Sukhocheva).

13

Let A and V be the same as in Proposition 6.17, and let .9 be a finite-dimensional

subspace. Then (2 C 9A, AY C ?) a (V C 9, v, VV C i) (cf. [III] ). Hint: Use Corollary 6.18. 14

15

Let V be a W-non-contractive operator, and let Vxo = exo, Vx, = ex, + xo with e I = 1. Then xo is an isotropic vector in Ker( V - eI). Hint: Apply the transformation Kt ` (I' * (') to VI te(VV) and use the results of Exercise I1 on §4. Prove that if A C B, where A, B E 2, then Kr(A) C Kr(B) and Kt ' (A) C KF ' (B) when (' ;d f.

Remarks and bibliographical indications on chapter 2 In §§1,4,5 the exposition is carried out at first in the most general form-for operators acting from one space .W1 into another 2 (T:,W, .,Y2), and only later is it made concrete for the case of operators acting in a single space. This

is the first time, apparently, that this has been done (at any rate, so systematically; cf. the monograph [V] and our survey [IV]) and mainly in the

interests of the theory of extensions of operators (see Chapter 5). In this connection it should be borne in mind when reading these notes and especially the bibliographical indications that in them, with rare exceptions, no account

154

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

is taken of generalizations (as compared with the primary sources) made in

the text to the case of operators T : Y,2. §1.1. Adjoint operators (relative to an indefinite metric) were first considered in the space IIx by Pontryagin [1], in J-spaces by I. Iokhvidov [5], [6] and by Langer [2], in G-spaces by I. Iokhvidov [12] and next by Azizov and 1. Iokhvidov [1]. §1.2. The idea of using the indefinite metric (1.4) applied to graphs of linear operators for discovering the properties of the operators themselves was first put forward and used by Phillips [1]. Later Shmul'yan [4], [5] and others developed it. Our exposition here follows Ritsner's monograph [4]. §1.3. The formula (1.10) was established by Azizov. For all the rest of the material see [III]. § 1.4. The Propositions 1.11 and 1.12 are 'folk-lore'. Sources of Theorem

1.13 can be found already in Pontryagin's article [1]. The formulation and proof given in the text are due to Azizov who used an idea of Langer's in [XVI]. Normal points are studied in [X]. Theorem 1.16 represents a certain development of Langer's results [2]. §1.5. The device presented here of passing to the factor-space ,W/,W° was first applied in [XV]; Theorem 1.17 we find essentially in Langer [9]. §2.1. W-dissipative operators were introduced in the book [VI]. ir-dissipative operators were first introduced in a particular case by Kuzhel' [5], [6]; they were studied in detail independently by Azizov [1], [4], [5], [8], and also by M. Krein and Langer [ 3 ] ; J-dissipative operators by Azizov [ 5 ], [ 8 ] E. Iokhvidov [ 1 ], Azizov and E. Iokhvidov [ 1 ] in which the main theorem

of this paragraph was established. A geoemtrical proof of the well-known Lemma 2.8 (see [VII]) is due to Azizov. §§2.2, 2.3. The Definition 2.10 and Theorem 2.11 are taken from [X]. Corollaries 2.12 and 2.13 are due to Azizov, as is all the material in §2.3 (see Azizov [8] ).

§2.4. Theorem 2.20 has been borrowed from [VI] and [XXII]. The remaining material of this paragraph was obtained by Azizov. §2.5, §2.6. Theorem 2.26 due to Azizov generalizes the corresponding result of Pontryagin for 7r-Hermitian operators (see also [XIV], [XVI]). The other results of these paragraphs were also established by Azizov.

§3.1. In an abstract formulation (but in a different terminology) rHermitian and a-self-adjoint operators were first studied by Pontryagin [1], who mentions that his attention was drawn to them by S. L. Sobolev (see the

remark on Chapter 3 below). After Pontryagin they were studied by 1. Iokhvidov [1], [2] (in the 'finite-dimensional' case cf. Potapov [1] ), and in more detail see [XIV]. J-self-adjoint operators were considered by Ginzburg [2], I. Iokhvidov [6] and later (in great detail) by Langer [1]-[3]. G-symmetric and G-self-adjoint operators are encountered partially even in Langer [2], and they are considered in detail in [III], to which we refer the reader for details. Proposition 3.7 and its Corollaries 3.8, 3.9 are found in an article by Azizov

Remarks and bibliographical indications on Chapter 2

155

and E. Iokhvidov [1]. As regards Corollary 3.12 see I. Iokhvidov [6], Langer

[2]. Remark 3.13 is due to Azizov, Corollaries 3.14 and 3.15 go back to Pontryagin [1] (cf. [XIV]). §3.2. In proving Theorem 3.19 it would have been possible to use Proposition 3.7 and to refer to a well-known `definite' result. However, we decided to bring in what we think is a simple proof, due to Azizov. Lemmas 3.20 and 3.21 used in it have such a tangled pre-history that we are inclined to attribute them to 'folk-lore'. Lemma 3.22 is due to Azizov. Corollary 3.25 is found, essentially, in Potapov [1]. Theorem 3.27 in the case of a continuous operator A is due to Ginzburg [2], and in the form in which it is formulated-to Langer [8]. §3.3. All the examples, except Example 3.33, are due to Azizov. §4.1. Plus-operators Vin a real space II, (9) v = II,, V bounded) were first

considered by M. Krein (see M. Krein and Rutman [1], and later in an arbitrary (complex) II, by Brodskiy [1]). The latter, in contrast to our Definition 4.1, imposed the condition ' v = Ilk (we point out that from this condition v n t,+ + = 0 already follows-see Lemma 1.9.5). The definition and the name 'plus-operator' of a plus-operator V in a general J-space itself were introduced in the articles of M. Krein and Shmul'yan [1], [2]. In contrast to Brodskiy, with these authors 9 v = . and the operator V is bounded a priori. For such operators they formulated Theorem 4.3 and presented for it a proof which remains valid in the more general case (see [XVI] for the previous history of this theorem), and they established a classification (which had appeared earlier in a particular case in Brodskiy's [I] ) of plus-operators into strict and non-strict (this terminology itself is due to them).

J-non-contractive (J-non-expansive) operators in a Krein space were intro-

duced and studied by Ginzburg [1], [2], generalizing the corresponding considerations in Potapov's [1] for finite-dimensional spaces. Earlier M. Krein [4] (see also [XIV]) had considered in IIk the so-called non-decreasing linear

operators V: L4 v= Ilk, V bounded and [Vx, Vx] > [x, x] for XE

.?+. We

point out that this inequality means, even in the general conditions of Definition 4.1, that V is a strict plus-operator with µ+(V) > 1 (cf. [XVI]). For plus-operators V in IIk with v = II, Brodskiy [ 1 ] discovered that they are either finite-dimensional (and then, possibly, unbounded), or they are continuous (cf. with our Corollary 4.8). As I. Iokhvidov [17] pointed out, the same Brodskiy article contains essentially all the ideas used in the proof of Theorem 4.6 and its corollaries. For these results in a rather fuller and explicit form, see I. Iokhvidov [17]. T. Ya. Azizov pointed out that the requirement that , v n ,Y+ * 0 imposed in these papers can be omitted in Theorem 4.6. Regarding Corollary 4.7 see I. Iokhvidov [17]. A curious generalization of plus-operators V was recently proposed and investigated in [XVI]: [ Vx, Vx] > µ [x, x] for some µ E fR and all x E H. For such V finite µ±(V) again exist and µ_ (V) 5 µ < µ+(V), and a number of facts were established, many of which probably remain true in the more

156

2 Fundamental Classes of Operators in Spaces with an Indefinite Metric

general situation when V:. '1 -.02 (in the spirit of our Definitions 4.1 and 4.2). §4.2. The concept of a doubly strict plus-operator and the basic facts about such operators in J-spaces were established in the articles of M. Krein and

Shmul'yan [2), [3]; there is a later bibliography of this topic in [IV]. The proof of Proposition 4.10 in the text was given by I. Iokhvidov [17] (there the priority of Yu. P. Ginzburg on this question is pointed out). J-bi-non-contractive (J-bi-non-expansive) operators were introduced and studied by Ginzburg [2]. Theorem 4.19 and its Corollary 4.20 are due to M. Krein and Shmul'yan [2], but the proof in the text is Azizov's (cf. Ritsner [4] ). It embodies most of the earlier known criteria for J-non-contractive operators V to be J-bi-non-contractive (see Ginzburg [1], [2], I. Iokhvidov [14], [17],

M. Krein and Shmul'yan [1], [2]; in connection with the rejection of the a priori requirement for V to be continuous, see I. Iokhvidov [ 17]. See Ritzer [4], E. Iokhvidov [8] for the generalization of the concept of a J-bi-non-expansive operator (cf. Shmul'yan [4] ). §4.3 Focusing plus-operators were first examined by Krasnosel'skry and A. Sobolev [1], later by A. Sobolev and Khatsevich [1], [2], and in detail by Khatskevich [6], [7], [10], [11], [15]; uniformly J-expansive operators-in the book [VI]. Theorem 4.24 was established by Azizov [10] (see Azizov and Khoroshavin [1]). Some of its assertions were obtained independently by M. Krein and Shmul'yan [5]. Theorem 4.27 is contained essentially in [VI]. §4.4. Theorem 4.30 in the 'finite-dimensional' case was established by Potapov [1], in the general case-see Ginzburg [2], M. Krein and Shmul'yan [3]; the proof in the text is due to Azizov. Theorem 4.31 has been borrowed from [VI]; Corollary 4.32 is due to A. Sobolev and Khatskevich [1], [2] §4, Exercise 12. In connection with Remark 4.29 a warning must be given

against mechanical transfer of results of the type in Exercise 12 from plus-operators to minus-operators (respectively, from (J1, J2)-non-contractive to (J,, J2)-non-expansive operators); it must be remembered that the roles of

the subspaces .; ands (respectively, of the projectors P; and P,.-), i = 1, 2, are interchanged. §4, Exercise 26. B-plus-operators in space H were introduced by Brodskiy [1] (see [XVI] ), together with the name `B-plus-operator' itself. In conclusion we point out to the reader that much more information about the operators acting in II spaces that are mentioned in §4 can be found in the monograph [XVI].

§5.1. Isometric (in particular, unitary) operators in infinite-dimensional spaces with an indefinite metric were first considered by M. Krein (see M. Krein

and Rutman [1]), I. Iokhvidov [1], [2], I. Iokhvidov and M. Krein [XIV], [XV]. Proposition 5.2, see M. Krein and Shmul'yan [2]. Example 5.5 in the text was given by Azizov; for the other examples, see I. Iokhvidov [12]. Theorem 5.6 is due to I. Iokhvidov [12]. §5.2. In connexion with Theorem 5.10 see [XIV], M. Krein and Shmul'yan [3], and also cf. Azizov [11]. Corollary 5.13 is found in M. Krein [5].

Remarks and bibliographical indications on Chapter 2

157

§5.3. Theorem 5.18 (and its corollaries) in the case of a commutative group is due to Phillips [3]. In the general case of amenable groups we are inclined to attribute it to 'folk-lore', since the proof given in the text in no way corresponds to that given in [VI], for example, for a single operator. This fact

was formally noted by Azizov and Shmul'yan; see also Exercise 7 on this section. For groups in IIj this result was proved by Shmul'yan without the requirement of amenability. Definitions 5.21 and 5.22 were given by M. Krein [XVII]. Theorem 5.23 is

due to M. Krein [XVII] (for details, see [VII), the proof given here was somewhat modified by Azizov. For similar results for stable and strongly stable J-self-adjoint operators, see Langer [11, McEnnis [1].

§5.4. All the results of this paragraph are in M. Krein and Shmul'yan [3]. §5, Exercises. The operators in Exercise 4 are called J-unitary operators by Shul'man. The condition in Exercise 9 was first considered by Masuda [1].

§6. I. Iokhvidov [1] was the first to apply Cayley-Neumann transformations to operators in spaces with an indefinite matric. Then these transformations were considered in detail in [XIV], [III], and were widely applied by many authors. In §§6.1-6.3 we mainly follow Ritsner [4].

INVARIANT SEMIDEFINITE SUBSPACES

3

In this chapter we shall set out results on one of the central problems in the

theory of operators in Krein spaces and, in particular, in Pontryagin spaces-the problem of the existence of maximal semi-definite invariant subspaces for operators and sets of operators acting in these spaces. It will be assumed that the J-non-contractive operators appearing here are bounded and defined on the whole space.

Statement of the problems

§1 1

We have already encountered the concept of an invariant subspace in

Proposition 2.1.11. We now go into it in detail. Definition 1.1: Let T.

-

be an operator densely defined in a Hilbert

Y. We shall say that the subspace 2' is invariant relative to T if Jr FY = 2' and T: 2' - Y. In particular, the subspace 2' = (0) is invariant

space

relative to any operator T; in this case we put by definition p(T j (0)) = C. We note that if Tis an operator defined everywhere in .0, then the condition is always satisfied and moreover Cr -Y) - It would be IT _ possible in Definition 1.1 to require, instead of the condition that clrfl2' be

I:

dense in 2', the inclusion 2' C 9r, but then the set of invariant subspaces would be impoverished. On the other hand, it would be possible to extend this

class by dropping the condition that C4T r) 2' = 2' and leaving only the condition that it fl 2' C .', but then this would lead to the situation, unnatural in our opinion, when any subspace which intersected )T only along the

vector 0 would be an invariant subspace for T. We therefore stay with the Definition 1.1.

In Pontryagin's foundation-laying work [1] it is proved that (in the terminology we have adopted) every 7r-selfadjoint operator A in IIX has a x-dimensional non-negative invariant subspace 2+, which can be chosen so 158

§1 Statement of the problems

159

that Im a(A 12+) >, 0 (Im a(A 12) 5 0). Further development of this result led to the following problems. Problem 1.2: Does every closed J-dissipative (and, in particular, every J-selfadjoint) operator A have an invariant subspace 2+ E _lt+? If it does, is there an .T+ such that Im a(A I Y+) 3 0 (and for a J-selfadjoint A is there also one such that Im a(A I Y+) S 0)?

Problem 1.3: Let d= (A) be the set of maximal J-dissipative operators A with p(A) n C+ ;e 0 whose resolvents commute in pairs, and let Y+ (C.+) be their common invariant subspace, and let p (A 12'+) n C+ ;'6 0 (A Ed).

have a common invariant subspace k+ E /u+ which Does the family contains Y+ and is such that p(A I - +) n C+ ;6 0 (A E d)? We notice at once that, in such a general formulation, Problem 1.2 has a negative answer even for J-selfadjoint J-positive operators (see Theorem 4.1.10 below).

Example 1.4:

Let W1 = Lin (e) (j W; be a Hilbert space, with II a II = 1,

dim .i = oo, and e 1 Wi We define relative to this decomposition a .

completely continuous selfadjoint operator G in el by means of the matrix G22=aG'22, G=IIGGj1I?i=i, where G11=0, G2'2 is a negative completely continuous operator in . I, f E Xi \RGI II f II = 1, and a > 0 is such that II G II < 1, and we introduce in Ml the G-metric [x, y] = (Gx, y). Since r1 is decomposed into the sum of the neutral one-dimensional subspace Lin (e) and the negative AeI, and since it follows from the inclusion f E.I \,G,, that 0 o ap(G), so Ye1 is a G(')-space with x = 1 (see 1.§9.6). Let P be the orthoprojector from

1 onto .01'. Then PG is

a completely continuous G-selfadjoint operator, which has, as is easily verified, not one non-negative eigenvector. In the space

,N' = 1

O+

2,

W2=

(1.1)

1,

we define a J-metric [(x1,X2),(Y1,Y2)1 _ [J(x1,X2),(y1,Y2)] by means of the operator J (see 1.(3.9)) '1=II JuII%i=1,

Jll=G, J12=(I-G2)1/2,

Let Al be the orthoprojector from .

J21=J2,J22=-G.

on to .,Y1 (C.W), and let the linear

A2 I .'2 = I G I -'. Then A 1 = A 1* > 0, operator A2 be such that Ker ..d2 = and A2 = Az > 0. Since the operator Al is bounded, it follows (see 2.Proposi-

tion 1.9 and 2.Corollary 3.8) that A = Al J+ JA2 = A`30. Since W1 C VA,

we can express A in matrix form relative to the decomposition (1.1): A = II Aii II Ii=1 It can be verified immediately that

A11=PG,

A22= -GI GI-'

A12=P(I-G2)1i2+(I-G2)"21GI-1,

A21=0,

3 Invariant Semi-definite Subspaces

160

is well-known that the selfadjointness of G implies iI G I = c, and

It

G. From the matrix representation of the operator A it follows that 0 O ap(A), i.e., A > 0. Since A 21 = 0 and G I G I -1 eRG = eG, so A 'A C 9A. From this the equality VA" = CAA (n = 1, 2, 3, ...) follows. Therefore if X'+ (E,,#' ) were an invariant subspace relative to A, it would also be invariant relative to therefore CAA = W1 Q+

A 2 = 11 (A 2)tj II a=,:

(A2),1 =(PG)2, (A2)21=0,

(A2)22= PGP(I- G2)1,12-(I- G2)1/2G-1,

(A2)22=I2.

Therefore

if x = (x,, x2) E f+ (xi E .wi, i = 1, 2), then also

A2x= <(PG)2x, + PGP(I- G2)1U2x2- (I- G2)1/2 G- 'x2, x2) E Y+. Since

Ker(A2-I)= (<(I- G2)"2x2i -Gx2) 1 x2E

2)= J

2,

Ker(A 2 - I) does not contain maximal non-negative subspaces. Therefore Y+

Ker(A2 - I) also. Since (A2 - I)JA C J,, so (A2 - I)(Y+ fl 9A) C .0, also. From the relations fly -qA = Y+ and Y+ Ker(A 2 - I) we conclude that (A2 - I)(i+ fl gA) * (B). Since (A2 - I)(2+ fl 9A) C Y+ and all the non-negative subspaces in ., are one-dimensional, there is in, a non-negative eigenvector of the operator A and this will also be an eigenvector

for PG; we have obtained a contradiction, because PG has no non-negative eigenvectors.

We note that for the operator A in the above example p(A)=0 and therefore (see 2.Theorem 2.9) there are no maximal uniformly positive subspaces in !2A. At the same time, in Pontryagin's theorem mentioned above, since A = II., the domain !2A does, by I.Lemma9.5, contain such a subspace. It is therefore expedient to introduce the following

Definition 1.5: We shall say that a set of operators of = (A) in a Krein space = + O+ . satisfies condition (L) and we shall write ..el E (L) if, for any

A E V, there is contained in 9A at least one maximal uniformly positive subspace Y+ (in the general case, different _T+ for different A E /); in particular, if z( should consist of a single operator A, then we would write A E (L) and would assume that ,W C Q)A.

If in Problems 1.2 and 1.3 the operators A satisfy condition (L), then naturally, as we shall see later in Theorem 1.13, the non-negative invariant subspaces are to be sought in the lineal CAA itself. In such a formulation the Problems 1.2 and 1.3 have not yet been solved. Later we shall introduce additional conditions under which they have affirmative solutions.

§1 Statement of the problems

161

Another group of problems-for J-non-contractive operators V with 9v=.W-finds its origin in M. Krein's works [4], [5].

2

Problem 1.6: Does every J-non-contractive operator V have an invariant subspace 99+ E AN+? If yes, then is there an Y+ such that I a(V I Y+) I >1 1? Problem 1.7: Let W = { U) be a commutative family of J-non-contractive operators, and let Y+ be their common invariant non-negative subspace (in brief,

Q/

+ C'+. Is there an

§e+ Eilf+

such

that Y+ C 2+ and

w2+ C 9+? We shall show that Problem 1.7 is such a general formulation has a negative solution even for a family 4! consisting of a single operator U. Example 1.8: Let ,e = W+ O+ .e- be a J-space, and let [e, )O', [ei )i be orthonormalized bases in W+ and *,- respectively. We define on the basis

{ ei )o U { e,- ) i a J-semi-unitary operator U: U e o = eo + el + el ,

Ue=e,+1

(i = 1, 2, ...).

The subspace 91+ = C Lin [ e; ) i (C ,2+) is uniformly positive and invariant

relative to U; moreover it does not admit extension into a maximal nonnegative subspace invariant relative to U. For, suppose Y+ C 2'+ (E,/#+) and U9+ C 2+. Since P+Jv7+ = .1+ (see 1.Theorem 4.5), there must be in se+

a vector of the form eo + x- with x- ;d 0, x- E W-, and together with it the vector U(eo + x-) = eo + ei + ei + Ux-. Since ei E.'+, so eo + xi + Ux- E Y+ and (eo + x-) - (eo + el + Ux-) E 9+. Since 9?+ is non-negative we have ei = (I - u)x-, which, as is easily seen, is impossible; we have obtained a contradiction. We note (cf. Theorem 2.8 and Remark 2.4 below) that in this example the `corner' U12 (= P+ UP- I --) = 0.

Definition 1.9: A subspace . is said to be completely invariant relative to a family of operators J E [ T) if' C 1r and Tie = 2' for all T E J (in this case we shall also write JSe = 2'). To conclude this paragraph we point out that, if .'+ is assumed to be a completely invariant subspace relative to ?l, then the solutions of the problems 1.6 and 1.7 have not yet been found. Later, in §§2-5, we shall indicate sufficient conditions under which these problems have an affirmative solution.

3

Finally, the third of the main directions is that generated by Phillips's paper

[3].

Problem 1.10: Let .4 = (A I be an algebra of continuous operators acting in the whole of a Hilbert J-space W; let art be closed relative to J-conjugation,

3 Invariant Semi-definite Subspaces

162

i.e., A E 4 - A` E . I, and let .1 contain I. Under what conditions on W will every dual pair (2+,.'-) which is invariant relative to -d (i.e., C 2', 2'+ C °-, 2+ [1] 2-) admit extension into an invariant maximal dual pair? Is commutativity of got sufficient for this condition?

Problem 1.11: Under what conditions on a group W = ( U) of J-unitary operators will every dual pair (2+, 2-) which is invariant relative to Pl admit extension into a maximal dual pair invariant relative to Pt? Is commutativity of P/ sufficient for this condition? Remark 1.12: In Problem 1.11 the invariance of the subspaces Y± implies their complete invariance, because Pt is a group and therefore it follows from

U2+ C 2+ and U-'2± C 2+ that U2+ = 2±. Moreover, if (2+, .'-) is an invariant dual pair of any J-unitary operator U, then it is easy to see that (C Lin (U-"9?+)o, C Lin (U-"2- )o) is a dual pair which is completely invariant relative to U and which extends (2+, 2- ), and even more precisely, it enters into any extension of (2+, 2-) into a completely invariant dual pair. We point out, regarding their above formulations, that Problems 1.10 and 1.11 have not yet been solved for the case of arbitrary commutative algebras and groups respectively. In §§4 and 5 below we shall give some sufficient conditions for them to be soluble affirmatively (see also 2.Theorem 5.18). We observe also that, under the conditions of Problem 1.10, the algebra .4 coincides with the linear envelope of its J-self-adjoint elements A (A`) with 11 A 11 < 1, and therefore it is contained in the linear envelope of J-unitary operators

U=(I-A2)in+iA, where

(I-A2)"i2= - 1

27r,

ro

]X(I-A2- XI)-1 dx,

the contour F. contains a(I - A 2) within itself and is symmetric about the real axis and lies in the open right-hand half-plane, and Re , X > 0. If now 2' is a maximal semi-definite invariant subspace of the operator U, then UY = 2' (see 2. Remark 4.21), and therefore U-'2'= Y. Consequently

AY=-

(U- U`)2'= 2i (U- U-')2'C Y.

Conversely, if AY C 2', then it follows from the definition of the operator !-A 2) in that (I -A 2)'/22' C 2', and therefore U2' C 2', and, by virtue of 2.Remark 4.21, U2'= Y. Thus, this Remark 1.12 enables us to go into the investigation of Problem 1.11 only, we leave the reader to make the appropriate reformulation for the Problem 1.10 himself.

4

In this paragraph we point out a connection between Problems 1.2, 1.3 and

1.6, 1.7 respectively.

§1 Statement of the problems

163

Theorem 1.13: Let A E (L) be a maximal J-dissipative operator. Then there is a E p(A) fl C+ such that, for 2+ E _tl+, the following conditions are equivalent:

point

a) 2+ C 9A and AY+ C 2+; where UU = Kr(A) = (A - f1)(A - CI)-' b) UY+ C

is the (see 2. §6.4).

9+,

Cagley-Neyman transform of the operator A at the point

Since A E (L), we suppose (see Definition 1.5) that

C f 1A, and with Im > 211 AP+ II belong to p(A) (see 2.Theorem 2.9). Let 2+ C FDA, 2+ E ail+, and AY+ C 27+. Then all the (' specified above 1W+

therefore all

12+, and since A 12+ is belong to the field of regularity of the operator A continuous, they are regular for this operator. Therefore (A-('I)-'Y+_Y+and

Thus, for the implication a) - b) to hold, it is sufficient to take Imp'>2IIAP+II.

with

We shall now prove that among these there are some for which the reverse implication b) - a) holds. Let 27+ E it(+ and Ur2+ C 2+. It is easy to see (see 2.Proposition 6.16) that a) will follow from b) if and only if (Ur - 1)9?+ = Y+ Let K be the angular operator of the subspace 27+. Since (Ur - I )2+ C 2+,

coincidence of these lineals is equivalent to maximal non-negativity of (see l.Theorem 4.5) (Ur - I)2+, or to the equality + = P+ (Ur - I) (P+ + K),Y+, which, in turn, is equivalent to 0 E p(Urji - I+ + Ur12K), where U-,j (i, j= 1, 2) are the blocks of the operator Ur (see 2.(2.1)). Since it follows from 2.Lemma 2.8, 2.Theorem 2.9, and 2.Formula (6.36) that

Ur12= -(Urii-I+)Ai2(A22-(I so

ur11 - I+ + Ur12K= (Ur11 - I+)(I+ - A12(A22 - I )-'K), and therefore it is sufficient for us to verify that 1 E p (ur11) and it is possible to choose with Im > 211 AP+ II so that II A 12(A22 - ')'II < 1. Using

2.Formula(6.37) and supposing without loss of generality that E p(A 11) we have

(Ur11 - I+)(I+ - A12)(A22 - I )-'A21(A11 - I+)-') =

f)(A11 - I+)-'

This equality implies that R,,;,, - r coincides with W+. Since Ur is a J bi-

non-contractive operator (see 2.Theorem 6.13), OEp(Uii),

II U111 II 5 I

(see 2.Remark 4.21), and therefore we have from I+ - Ut i = U11 (Ur - I+) that 9lr - u; = -W+, which implies, as is easily seen, the inclusion 1 E p (U1 i ), and therefore 1 E p (Ur11). with Im (' > 211 AP+ II We now show that there is a

II A12(A22 - I-)

11 < 1.

such that

164

3 Invariant Semi-definite Subspaces

Since the operator A is J-dissipative 0<

Im[A11x+, x+] + Im[A12x-,x+] + Im[A2ix+,x-] + Im[A22x-,x+]

for all x+M+, x- E.)e- fl 'A. In particular, on substituting instead of x the vector x. (() = Im"('(A22 - ("I-)-'x-, a E IR, for all x- E Y_ with II x II = 1 and +_

A12(A22 -

('I-)-1x-

II A12(A22 - ('I-)-'x-II

x

it is assumed that A 12 (A22 - ('I- )x- # B-for, II A12(A22 - rI-)-'x- II = 0 5 q < 1 for any q > 0), we obtain

(here

II A12(A22- ('I-)-'x- I <

Im[f111x+, x+] Im " ('

otherwise,

+ ImIA,,x+. (A» - tI-)-'x-l

+ Im"(' Im IA22IA22 - ('I_)_ IX-, (A22 - ('1 )-'x

]

From 2.Theorem 2.9 and 2.Corollary 2.8 we obtain that II 1/IM (', and when Im (' > II Re (II we have II (A22 II A22 (A22 - tI )-' II I + 2, and therefore II A2 1

All II II A12(A22 - r)-'x - II S IIIm"(' + Im

1 +2

II

+ Im

Since for 0 < a < 1 and sufficiently large Im r we have IIA11II+IIA21 1+ Im (' Im '(' Im

"r

and since x- (II x-1 II = 1) is arbitrary, we obtain

II A12(A22 - ('I-)-l II < I. 1.14 If A E (L), foA =.W, 27 E,41-, and A: 2- - 2-, then T- is an invariant subspace of the operator A. In accordance with Definition 1.1 it is only necessary to verify that

9A fl Y-= 2-. From 1.Corollary 8.16

it

follows that W+ + 2- = M.

Since ye+ C 9A, we have 9)A =,W++ (2- fl GA), and therefore

implies l 2 = 2-.

W

Theorem 1.13 and Proposition 1.14 show that in future we need investigate only Problems 1.6 and 1.7, and the reader himself will be able to formulate the corresponding consequences for maximal J-dissipative operators A E (L) or for families of them ,.d E (L), and so we shall now dwell on them in the main text. In conclusion we mention that, historically, solution of the Problems 1.6 and 1.7 did not always precede the investigation of Problems 1.2 and 1.3; thus,

for example, Pontryagin's theorem mentioned in §1.1 was obtained earlier than Corollary 2.9 set out below in §2, from which we suggest to the reader in Exercise 10 on §2 that he should derive it.

§1 Statement of the problems 5

165

In the preceding paragraphs of this § 1 we decided that we would investigate

the problem of invariant subspaces mainly for J-non-contractive operators and, in particular, for J-semi-unitary and J-unitary operators. For brevity in the further exposition we introduce

Definition 1.15: We shall say that a family W= (UJ of J-non-contractive operators has: a) property -1, + (respectively, 4 _, (D [ll) if it has a common invariant subspace 2'+ E ft+ (respectively, 2' E Jr, 22+ E Jt+, 2+ [1] 91-);

b) Property i+ (respectively,-,) if each of its common completely invariant non-negative (respectively, invariant non-positive, completely invariant non-negative and non-positive) subspaces (including also (0)) admits extension into a maximal semi-definite subspace (of the corresponding sign) invariant relative to q/; c) Property 4 [lJ if each of its invariant dual pairs (2+,-) with

U9?' = 2+ (u E q/) admits extension into a maximal dual pair invariant relative to W. It is clear from Definition 1.15 that if a family W has the properties ( +, c, W, then it also has the properties 4) ±, (D, 'i) respectively, and the property 4) (4)) is equivalent to having properties 4+ and i,_ (c+ and c _) simultaneously. We note also that Definition 1.15 is applicable, in particular, to a family q/ consisting of a single operator U.

To conclude this section we indicate a device which will enable us later to simplify the proof of whether certain J-non-contractive operators (or families L-I and 4' Li I , and to avoid of such operators) have the properties repetitions in the discussions. 6

Definition 1.16:

Let q/ = (U) be a set of J-bi-non-contractive operators. We shall say that a complex y' of properties of the set q/ (the complex may, in particular, consist of just one property) is 4-invariant relative to q! if this same complex of properties is also possesed by every 4/ = (UJ of J-bi-noncontractive operators acting in the factor-space . = 9901l/ 'o (see 2.(1.11)), where 'o (C,-.?O) is an arbitrary subspace completely invariant relative to >>/. It is well-known that (in our terminology) properties of families of

operators, such as the property of being a group, commutativity, etc., are also 4'-invariant. Moreover a proposition such as follows also holds: 1.17 Let -V/ = (U) be a family of J-bi-non-contractive operators. The property that 199 C .i°° and 2' C U91 implies UY C .-°1 is -,D -invariant, and if

W has this property, if 2'+ is a completely invariant subspace relative to /'+ C .:+, and A'0+ = 2'+ r11+1, then U2'°+ = Yo++.

P7/,

3 Invariant Semi-definite Subspaces

166

Let 2'o be an arbitrary neutral subspace completely invariant relative to ,!/, let

.

= 201] /2o, and let '! = (U) be a family of J-bi-non-contractive

operators generated by the family 4/. If SP is a neutral subspace in . if and if C U 2 ' for every CIE 4 / , then 2 = Lin (x E 1 1 2 E 2') is a neutral subspace in

.Y( and 2 C U. = (Ux E Ux I z E 2'). Then by hypothesis U2' is a neutral subspace, which implies that 02 (= U2') is neutral. Now let 2'+ C O+ and U2'+ = 2+. It follows from this that Yo+ C U2'°+ (C 2+ ). By hypothesis U2'°+ is a neutral subspace, and since it contains Yo +, so (see 1 Proposition 1.17 and 1. Definition 1.13) 4/2'° = 9?0

Definition 1.18: A semi-definite subspace is said to be a maximal invariant (respectively a maximal completely invariant) subspace for the family W if it does

not admit non-trivial semi-definite extensions of the same sign in the set of subspaces invariant (respectively completely invariant) relative to W. A maximal invariant dual pair is defined similarly. It follows directly from Zorn's lemma that 1.19 Every semi-definite invariant (respectively, completely invariant) subspace admits extension into a maximal invariant (respectively, completely invariant) semi-definite subspace of the same sign. A similar proposition also holds for invariant dual pairs').

Lemma 1.20: Let (,Yt°r) be the set of J-spaces, and let (Wj,,x = ( U)) be the set of all families of J-bi-non-contractive operators in which is inherent a

certain complex .Y( of -t-invariant properties, including the property (2' C ?°, 2' C U2 - U2' C 3°). Then either every such family has the property 4)+, or among them there is a family (q/) which does not have the property 4)+ and is such that each of its completely invariant non-negative subspaces is positive.

Suppose a family 4/ _ (U) satisfies the condition of the theorem and nevertheless does not have the property 4+. We assume that 2° is a maximal neutral subspace completely invariant relative to W (see definition 1.18). This enables us to pass to the family L/ _ (U) of J-bi-non-expansive operators acting in the f space . = 201120. In accordance with the condition, this family has the complex of properties .Y( and, by virtue of Exercise 21 on 1.§8, like 4/ it does not have the property 4+. We now show that all non-negative subspaces 2+ which are completely invariant relative to 4/ are positive. Let 2°+ be the isotropic part

of such a subspace .+. Then, by Proposition 1.17, .P°+ is a neutral subspace completely invariant relative to every operator U E 4/, and this implies that complete invariance of 2' = Lin (x E R I R E 2°+) relative to U E I. Since We draw the reader's attention to the fact that a maximal invariant semi-definite subspace is not

assumed to be, generally speaking, a maximal semi-definite subspace, and the concepts of a 'maximal invariant dual pair' and an 'invariant maximal dual pair' (cf. 1.DefinitionlO.l) are, in general, not identical.

§2 Invariant subspaces of a J-non-contractive operator

167

it follows from the maximality of 20 as a neutral subspace `1'o c 2 C completely invariant relative to W that 2'o coincides with 2, i.e., -_o+ = (0). Remark 1.21: Corresponding propositions about properties 4,-, 4,1-L1, can be proved in an entirely similar way (see Exercises 4-6).

Exercises and problems Construct a J-positive completely continuous operator A and verify that if the J-space is not a Pontryagin space, then A -1 has no invariant subspaces 2+ E .,ll' which are contained in CIA (Larionov [9] ).

1

2

Let 4V= (U) be a group of J-unitary operators. Prove that for it the properties: a) amenability; b) solubility; c) commutativity; d) UE Q/= U*E 4I; e) uniform boundedness; f) U` = U* = U-1 are 4)-invariant in the sense of Definition 1. lb. Hint: In proving d) first verify that under the conditions of Exercise 2 U* = U.

3

Let it be a family of J-bi-non-contractive operators. Verify that the property ( every

neutral invariant subspace of the family W admits extension into its completely invariant subspace) is 4?-invariant (Azizov). 4

Let Yl be a certain (D-invariant complex of properties inherent in families W/,.W

U)

of J-bi-non-contractive operators and containing the property the 4)-invariance of which was asserted in Exercise 3. Then either every such family has the property 4)_, or among them there is a family (V/) which does not have the property 4)_ and is such that each of its invariant non-positive subspaces is negative (Azizov). 5

Prove that under the conditions of Lemma 1.20 (respectively, of Exercise 4) either each family 4/j,.x there mentioned has the property 4)+ (respectively, $_), or among these families there is a family (a+/) which does not have this property and is such that it has a

maximal completely invariant non-negative subspace .W+ (respectively, a maximal invariant non-positive subspace Y-) which is definite and _T+ !t ..//+ (respectively,

ll.i/l ) (Azizov). 6

If under the conditions of Lemma 1.20 4+/j,,Nis a group of J-unitary operators, then either every such group has the property 4) Ill (respectively 4)1±]), or there is among them a group (-V/) which does not have this property and is such that all its maximal invariant dual pairs are definite (respectively, which has a maximal invariant definite dual pair (.V+, Y-), 99+ 0.,//±) (Azizov). Hint on Exercise 4-6. They are proved similarly to Lemma 1.20.

Invariant subspaces of a J-non-contractive operator

§2

Let.

= . + O+ W- be the canonical decomposition of a Krein space, and let Vi; 11 ?j=1 be the matrix representation (see 2.(2.1)) of a J-non-contractive operator V relative to this decomposition. 1

V= 11

Theorem 2.1: Every uniformly J-expansive operator V has the property 4)+. If V is a uniformly J-bi-expansive operator, then it has the property 4) and it has a

single pair 2+ C .O+ and 2- C .?- of maximal semi-definite invariant subspaces; moreover they are uniformly definite, I a(V I 2+) I µ+ (V) (> 1), and a ( V I 2 )I <, max(0; µ_(V)) (<1).

3 Invariant Semi-definite Subspaces

168

We begin with the assertion about a uniformly J-bi-expansive operator V. By 2.Theorem 4.13 the unit circle is free from the spectrum of the operator V, and therefore its Cayley transform A = K; ' (V) = i (V + I) (V - I)- I will, by virtue of 2.Proposition 6.16, be a bounded uniformly J-dissipative operator and a(A) fl (- oo, oo) = 0 (see 2.Proposition 2.32).

We write a± = o(A) fl C, and let r± be Jordan contours lying in C+- fl p(A) and surrounding a±. By 2.Theorem 2.21 the subspaces Y'+

= P+J( and 99- = P-LW, where

P±_- tat r. (A-XI)-'dX, ,

(2.1)

are respectively non-negative and non-positive invariant subspaces of the operator A. Since P+ + P_ = I (see 2.Theorem 2.20) Y+ + 9?- _ .W

(2.2)

and therefore (see 1.Proposition 1.25) Y± E ill±. From (2.2) and the invariance of 2 relative to A it follows immediately that Y± are also invariant relative to V(= K, (A) = (A + iI) (A - iI) -' ). Since Visa uniformly J-bi-expansive operator, we have: a) An arbitrary vector y E 2+ is, by 2.Theorem 4.17, expressible in the form y = Vx (x E Y+ ), and by 2.Theorem 4.23 there is a S > 0 such that

[y,y] > [x,x]+6llxl12> III IIYIIZ, i.e. Y+ is a uniformly positive subspace; b) For an arbitrary x E 9?- the inequality - [x, x] > - [ Vx, Vx] + 6 II X II 2 > S 11 X II 2 holds (since Vx E : - ), and therefore Y- is a uniformly negative subspace.

Therefore Y` are Hilbert spaces relative to the scalar products ± [x, y] (x, y E Y±), and so in them V I Y+ is an expansive operator and V I 97 is contractive, which implies the inequalities I a( V I p7+) I> 1 and I a( V Y-) 11. But since (see 2.Remark 4.25) any operator X V with 1

µ+(V)

<X<

1

max[O; µ(V))

such that I a(X M Y+) I > 1 and I a(X M Y-) I< 1 will be uniformly inequalities and I a(V I 2+) I > 'µ+ (V) the J-bi-expansive

I a(VIY-) I <, max[O; µ_(V) also follow. Now let the subspaces Y+ (C ?±) be such that V2+ = 2+ and V22_ C Y_. Just as above we can verify that Y+ are uniformly definite, and moreover Ia(VIY+)I>,µ+(V) and Ia(VI'_)I<,max10;A-(V)). From the properties of the Riesz projectors (2.1) (see 2.Theorem 2.20,e)) we obtain Y'± C 91±, and therefore the operator V has the properties 4)±, i.e., the

§2 Invariant subspaces of a J-non-contractive operator

169

property 4 . At the same time we have verified the uniqueness of the maximal semi-definite invariant subspaces of the operator V.

Now let V be an arbitrary bounded uniformly J-expansive operator. We suppose that we have succeeded in extending the J-space .3r into a J-space .Ye = We+ O+

(J = I+ O+ J), and the operator V into a uniformly J-bi-

expansive operator V. Then, by what has been proved above, the operator 17 has a unique maximal uniformly positive invariant subspace 99+ = Lin (z+ + K9+ 19+ E'+ 0 + ) (K E X (M+ 0 W+, tee- ), 11 K 11 < 1), and therefore Y+ (=.r, n W= Lin (x+ + Kx+ I x+ E ..e+)) E Ill+ (M) and V,-W+ C 9?+. Now, if Y+ C .40 +(,W) and V'+ ='+, then V?+ = T+ also.

Hence, 2'+ c k+ n .re = 2+, which proves that the operator V has the property '1+. Therefore to complete the proof it remains for us to construct the extensions mentioned above of the space X and the operator V.

Let V be a uniformly J-expansive operator. Then A = V` V -I is a uniformly J-positive operator (see 2.Theorem 4.24). By 2.Corollary 3.29 the

operator A has uniformly definite invariant subspaces 2i E _lt±, and Y1 [1] Y1, and so without loss of generality (see 1.§7.6) we shall suppose 9?1 = re+- . Since (A, ' C M±) is equivalent to (V ` V WC W± ), we have VM+ [1] V.W-. Since V is a J-non-contractive operator, so (see 2.Corollary 4.13) V.ye+ is a uniformly positive subspace with the angular operator V22 V111 defined on , v,, and 11 V21 VIi 11 < 1. Let I' be the extension of the operator V22 Vi 1' on to the whole of JY+ and let II F II < 1.

We construct a J-unitary operator U(T) by the formula 2.(5.3). Then

V = U(r) W, where W = U-' (F) V is a uniformly J-expansive operator, and WMe+ C ye+ and W.le+ [1] Wye-. Now as -W+ we take one more copy of .-W+ and we extend the J-unitary operator U(r) into a J-unitary operator U(P) in . e = .re+ O+ .ye by putting r D IF, P .;e+ = 0, i.e., U(P) .)e+ = I+. After this it is sufficient now to construct an extension of the operator W into a uniformly operator defined on Let J bi-expansive f iP. [ Wx, Wx] [x, x] = S 11 x 11 2, S > 0. We define the operator W I ie+ by putting Wx+ = Woox+ + W1ox+ + W2ox+, where x+ E)'e+ and

woo = Wii: J°+ -.Ye+; W10 = ((1 + a)I+ + W12 W2)U2U1o:

+_

"y+,

U,o is a partially isometric operator mapping .ye+ on to Ker W11 (C.W+) and Ker U1o = W20 = W22 W*12((1 + CO I+ + W12W1*2 )-1i2U1o: ,y+ _.O-.

It is immediately verifiable that, when 0 < a < S/(l + 11 W22 112), the operator will be uniformly J-bi-expansive. ,

Let T= 11 Ti; 11 ?;-, be the matrix representation of an operator T, defined everywhere, relative to the canonical decomposition -e = W+ O+ .-e- of the 2

3 Invariant Semi-definite Subspaces

170

J-space .. We bring into consideration the functions

Gi(K+)=K+ T,1+K+T12K+- T21- T22K+,

(2.3)

Gr (K-) = K_ T22 + K_ T21K_ - T21 - T,1K_ ,

(2.4)

whose domains of definition are respectively the operator balls X1 (see 1.8.19).

Lemma 2.2: A subspace Y+ E -tf± with angular operator K± is invariant reative to an operator T with )T= W if and only if the operator K+ is a solution of the equation Gf (K±) = 0 respectively.

Let TY+ C Y+ or, what is equivalent, suppose that for every x+ E W+ + which is a solution of the equation

there is a y+ E

T(x+ + K+x+) = y+ + K+y+), which, in its turn, is equivalent to the system TI2K+x+ = y+ T21x+ + T22K+x+ = K+y+}

(2.5)

We now substitute the value of y+ from the first equation of the system (2.5) into the second and, since x+ is arbitrary in ,.+, we obtain that Gi (K+) = 0. Conversely, suppose Gi (K+) = 0. Then we take as the required y+ the vector defined by the first equation of the system (2.5).

The assertion that TY- C £- a Gr (K_) = 0 is proved similarly.

Definition 2.3: We shall say that an operator T satisfies the condition A+ (A-) and we shall write T E A+ (TEA-) if there is an operator K+ E X +, II K+ II < 1 (K_ E .yl-, II K-1 I < 1) such that Gi (K+) (respectively,

Gr (K_) is a completely continuous operator. Remark 2.4: Using 1.Theorem 8.17 we can by a simple calculation satisfy ourselves that T E A+ (respectively, T E A_) if and only if there is a canonical decomposition ,Y = W' [ + ] ,- (respectively, .0 = R' [ + ],,912- ) such that the `corner' Pi TP1 I i (respectively, Pz TPz I ' ), where P;± are the J-orthogonal projectors on to (i = 1, 2) is completely continuous. Therefore the inclusions T E A+ or T E A_ do not depend on the actual decomposi-

tions of the space as might at first sight appear from the Definition 2.3. It follows from Theorem 2.1 that every uniformly J-bi-expansive operator satisfies the conditions A+ and A_. However, even for such operators there is not always a single decomposition for which both the `corners' are completely continuous simultaneously, or, what is equivalent, there is not a Ko EX' with II Ko II < 1 such that Gi (Ko) E .y'., and GT (Ko) E / O simultaneously (see Exercise 2 below). Nevertheless the following proposition holds:

§2 Invariant subspaces of a J-non-con tractive operator

If U is

2.5

a

J-bi-non-contractive J-semi-unitary

U E A_ ).

{ U E A+)

operator,

171

then

In particular, if U is a J-unitary operator, then

{UEA+) (UEA-). Let U21 E .y'.. It follows from 2.Corollary 5.12 that U12 E .Se., and hence

(UE A-). The assertion about J-unitary operators is proved similarly.

3

Let V be a J-bi-non-contractive operator. We introduce the notation

1+(V)=

V-T+=-V+,I a(VI1+)I %:µ+(V)),

(2.6) (2.7)

IVY- =Y, Ia(VIY )I<,max(o;µ (V)). -T (V) For the proof of Theorem 2.8 below we need the following simple proposition: 2.6 Let

+- E

and let K± respectively be their angular operators and

let !± be invariant relative to an operator T with VT= W.

Then

a(T 19+) = a(Tl l + T12K+) and a(T 12-) = a(T22 + R21K- ). This proposition is a corollary of the fact that the projectors P± map V'± homeomorphically on to +- (see 1.Theorem 4.5) and

TIY+=(P+I

2+)-'(T11+T12K+)P+I Y+

and

TI -T- =(PI

mo)-'(T22+T21K_)P-If-.

Moreover, we shall repeatedly use the following lemma which follows from, for example [VII], Theorem 4.5.6.

Let T be a completely continuous operator acting from a Hilbert space e1 into a Hilbert space ,02. If (K6) and ( F6) are bounded generalized sequences of operators consisting of operators acting from .2 into *1 and converging in the weak operator topology to Ko and Fo Lemma 2.7:

respectively (K6 - KO, Fb - FO), then ( TK6) converges in the strong operator topology to TKO

(TK,

(s)

'' TKo) and F6 TK6

Fo TKo.

Theorem 2.8: If a J-non-contractive operator VE A-, then it has the property If V is a J-bi-non-contractive operator, then

a) VEA+=,)±(C`);4 0 and T(V`);4 0; b) VEA- = f-(V`)= E /V I.,l._ =Y+111,x'+E,7+(V)) and VEA+-,q+(V`)=

(A'+E._tf+IA"+='-[1],

Y-E/-(V));

c) VEA-- Lin (X (V) II X I > 1)C fl(Y+I9+E/+ (V)) and VEA+>Lin(X (V)IXI <1)C fl(-T-191-E/-(V)).

Later, in Theorem 3.9, it will be proved by another method that V has the property i+.

3 Invariant Semi-definite Subspaces

172

Ll Let V be a J-non-contractive operator, and let I = (1 + en)P+ + (1 - en)P-, where 0 < c,, < 1. Then V = VI,, is a uniformly J-expansive

operator: [ VInX, VInx] > [Inx, InX] > [X, x] + en II X II

2.

Since Vn11 = (1 + e,,) V,1, the operator Vn will be uniformly J-bi-expansive if

and only if V is a J-bi-non-contractive operator (see 2.Remark 4.21). It follows from Theorem 2.1 that the operator Vn has an invariant subspace Y' E ..//+. Let Kn denote its angular operator. By Lemma 2.2 it satisfies the equation Gj,(K,,) = 0. Since (see 1.Proposition 8.20) the ball X+ is bicompact in the weak operator topology, we can choose from the sequence (K',} (c,, 0 as n - co) a sub-sequence (K,,) which converges in this topology to a certain operator K o E X + as n oo. Since e,, 0 when n co, KaV,,11 = (I + p,,)K,,V11 , Ko Vii, V2122K,1= (1 - e,t) V22

V,121 = (1 + ea) V21 - V21,

K,1-s V22Ko.

Since it follows from Remark 2.4 that we can without loss of generality assume that V12 E .99., we can use Lemma 2.7 and obtain KnV,112K,, = (1 - c,)K,,V12K;, - KO V12Ko.

These relations enable us to conclude that Ko satisfies the equation Gv (Ko) = 0, i.e., by Lemma 2.2 the subspace SP+ = (x+ + Kox+ I x+ E W+1 (Ea/+) is invariant relative to the operator V.

a) Let V be a J-bi-non-contractive operator. We verify that then I a(V I Y+) I > µ+(V). It suffices (taking Theorem 2.1 and 2.Remark 4.25 into account) for us to deal with the case when µ+ (V) = 1. By Proposition 2.6 we have p(V I 2P+) = p(Vo), where Vo = V11 + V1z_Ko. Since (see 2.Remark 4.21) 0 E p(V11) and V_T+ = Y+, so 0 E p(V, 1) n p(Vo). Since (see Exercise 4 on 2.§4, I = ( X II X I < 1) C p(V11) and V12 E .1 ,, it follows from 2.Theorem 2.11 that D C p(Vo). We _ shall prove that Y+ Ei+(V), i.e.,

ao = l n a(Vo) = 0. Let X E p(Vo) n OD. The operators V. =_ 17,111 + V,112K,, converge strongly, by Lemma 2.7, to the operator Vo, and by Theorem 2.1 and Proposition 2.6 we have X E p(V,,). We now verify that supn(II (V,, - XI+)-' II < oo. To do this it is, by a well-known Banach-Steinhaus theorem, sufficient to show that the set ((V,,- XI')-'x+} is bounded for every x+ E +. Let (V,, - XI+)-'x+ = xi We rewrite this equality in the equivalent form

[I+-e,1[(V0-XI +)-'V1zMI Ko)-I+1(170-XI +)

V1V12K,T)

I+)-'V12(Kn- Ko)]2}x,i _ [I+ - (V0- XI+)-'V12(K,1-Ko)](Vo- XI+)-'x+.

(2.8)

Since

II [(Vo-XI+) V12(Kn-Ko)]2II

52II

(Vo-XI+)-`V,z(K,,-Ko)(V0-XI+)

`V12I,

§2 Invariant subspaces of a J-non-contractive operator 0, and V12 E .v'm and by Lemma 2.7 (Vo - )'I+) -' V12 (K,r - KO)

K,1- K,,

173

(S) i 0,

so (see, for example, [XVIII] )

II(Vo- XI+)-'V12(Ka-Ko)(Vo-XI+)-'V2II -0. Sax+, where II T,, 11 - 0, and (S,1) Therefore (2.8) takes the form (I+ + TT)x,i = is a uniformly bounded sequence. Hence for sufficiently large values of >i we obtain that x i = I+ + T a ) ' S,,x+ and I xi) is a bounded sequence, and therefore (V# < Co. Now let Xo E co. Since ao is an isolated point of the spectrum of the operator I

Vo, there is an open disc Do (C D) whose boundary I'o consists of regular points of this operator and is such that Do fl ao = (Xo). Since Fo is compact we conclude that the set ((V,j - XI+ )-' I Xo E Fo) is uniformly bounded with respect to X and n, and therefore the sequence of Riesz projectors

Pa = - 1

2ai converges strongly to the projector

Po= -tai 1

ro

(Va

- XI+)-' dX

r, (V'o- XI+)-' dX

on the root space 2?(Vo) (see 2.Theorem 2.20). But Pa = 0 for all ri, and therefore Po = 0, which implies that 2'x0(Vo) = 10). Thus I a(Vo) I > 1, i.e., f+ (V) ;.d 0. Now let VE A+. We shall suppose that V21 E 91. (see Remark 2.4). We bring into consideration, by 2.Formula (5.3), the operator U(F) with F = V22 Vi1' (E .gym ). The operator U = U-' (F) V is, together with V, J-binon-contractive and, as is easily verified, II U22 II S 1. Therefore U22+F(I+_P*r)-1/2U12+((I-

V22 =

1,r *)_1/2-I-)U22

is the sum of a compression and a completely continuous operator. This enables us, using the same scheme as we used in proving that V E A_ - 9+ (V) # 0, i.e., again starting from Proposition 2.6 (but this time its second part), to prove that V E A+ = f - (V) * 0. We also notice the following obvious implication:

VEA± a V`EA-. Therefore, if V E A_ (respectively, A+ ), then

(2.9)

V`) ;4 0 (respectively,

1+(V) * 0). So assertion a) has been proved. b) Let 2+ E1+ (V). Then A"- = X+ I-'] E ..!!- and V`. IV- C ./I"-. Since I .,l _ -)= (V`)22 = V22 and (V`)21 = - V,Z, so by Proposition 2.26, a(V` u(V2*2 - V2Q), where Q is the angular operator of the subspace .'!"-. From V 2 E .9'm we conclude (see 2. Theorem 2.11) that [ X I I X I > 1) C )5( V` I ,I'- ). Let I Xo I > 1 and V`xo = Xoxo, 0 * xo E ./V-. Since moreover xo E :1'+ (see

2-Exercise 5 on §6), so xo E Poo and therefore xo E 2+ fl .4'-, which implies (see

3 Invariant Semi-definite Subspaces

174

2.Exercise 17 on §4) VV`xo = xo, i.e., Vxo = (l/Xo)xo and 11/X I < 1-we

have obtained a contradiction of the fact that Y+ Eq+(V). Therefore A"- E,J-(V`). Similarly it can be verified that, if /V_ E7_(V`), then 99+ = T- (-L] E '+ (V), and the second implication in b) holds. c) Let X E ap(V) with I X I ie 1 and let xE Y),(V). Then the spectrum of the restriction of the operator V on to the finite-dimensional invariant subspace

Lin ((V - XI)'x(o consists of the one point (X). By 2.Theorem 1.13 we have x [1] A' for all A' E(+ (V`) if I X I < 1 or for all .4' E j- (V`) if i X I > 1. Using the proposition b) which has already been proved we obtain that xE fl (2- 12- E f-(V)) (I X I < 1) or, respectively, that xE fl (Y+ I Y+ E f+(V)( (I X I > 1). Therefore the inclusions indicated in

assertion c) are valid for every 'x(V) and are therefore also valid for their linear envelope.

Corollary 2.9:

If V is a a-non-contractive operator or a 7r-bi-non-expansive

operator, then f±(V) ;4 0 and the inclusions in assertions b) and c) of Theorem 2.8 hold for V In particular, ir-semi-unitary and 7r-unitary operators have these properties.

This follows from the fact that V21 and V12 are finite-dimensional continuous operators and therefore V E A+ fl A_ . Corollary 2.10:

If V(EA+) is a J-semi-unitary J-bi-non-contractive operator,

then f±(V) ?6 0 and assertions b) and c) in Theorem 2.8 hold for it. In particular, J-unitary operators U E A+ fl A- have these properties. It is sufficient to compare Proposition 2.5 with Theorem 2.8.

Remark 2.11: In accordance with 2.Remark 4.29 all the statements of problems and results in §§1 and 2, as also in §§3-5 later, for J-non-contractive and J-bi-non-contractive operators can be reformulated without difficulty in terms of J-non-expansive and J-bi-non-expansive operators, and this we leave the reader to do.

Examples and problems 1

Give an example of a uniformly I-expansive operator which does not have the property & (and even less, the property 4_) (Azizov). ./P+, J4" (O( and in it Hint: Consider a J-space N' _ .,Y® ( ./P-, dim N1 = dim an operator X U where I X I > I and U is a semi-unitary operator mapping .rY into

2

Let.r = .,Y+ Q+ .YP- be a J-space, .,Y+ and ./P- being two copies of one and the same infinite-dimensional Hilbert space. Verify that the operator V = 11 V1 11 zJ= 1, where VII=AN, V12 = (1/2)I, V21=0, V22=,Y/2I, is a uniformly J-bi-expansive

operator. Prove that there is no Ko E .W + with IlKo ll < 1 such that Gi (Ko) E s/' and Gv (Ko*) E Y. simultaneously (Azizov).

§2 Invariant subspaces of a J-non-contractive operator 3

175

Let V be a J-bi-non-contractive operator with V E A_ (A+ ), let co be the spectral set of

the operator V with ao c (X II X I > 1) (respectively, ao C ED), and let PaO be the

corresponding Riesz projector. Prove that then PPO.f c fl (7+ )+ E/+( V) (respectively, P,,Yf c fl (2- I J'- E f - ( V)) (Azizov). Hint: Use 2.Theorem 1.13 and Theorem 2.8.

Let V be a r-non-contractive operator, let I Xo I > 1, and let 2'o be the isotropic

4

part of the lineal 91a0 '(V) (the possibility that 2',,O(V) = J'x 1(V) = J'o = (B)) is not

excluded). Verify that then dim 22o < dim 22x0(V) and that the subspace 2'o + J'xO( V) is non-degenerate (Azizov).

Hint: Use Theorem 2.8, 2.Theorem 1.13, and 2.Exercise 17 on §4. 5

Let V be a r-non-contractive operator, and let 22i, 222 be arbitrary invariant subspaces of it from .,//+. Prove that then, if I X I > 1, we have

dim(221,(VI2',)+2'i,-'(VIY',))=dim(2,(VI222+. x-'(V IX2))=dim 4(V), and if I X I = 1, then dim 22X( V I I',) = dim 2'a (V 12'2) (Azizov). Hint: Use Exercise 4, Theorem 2.8, 2.Theorem 1.13, and 2.Exercise 17 on §4. 6

Let V be a r-semi-unitary operator in 11 let .J'+ E /+ (V) and to (V I d'+) I > 1, and let ap (V) fl a *-, (V 197+) = 0. Then the operator V' has a single x-dimensional positive, and a single x-dimensional neutral, invariant subspace. When x = 1 the operator V` has no other invariant subspaces from //+, and when x > 1 the power

of the set of them is either not greater than 2,-2 or is not less than that of the continuum (Azizov). Hint: Use Exercise 6 and the hint for it. 7

Let II1 =11+ (D rl_ with 11+ = Linle') and let (e, 11' be an orthonormal basis in r1-. Prove that the linear operator V defined on e+ U (e, )i as Ve+ = 2e+ + ei , Ve; = e;+, (i = 1, 2, ...) is bounded and that its closure is a r-semi-unitary operator which satisfies the conditions of Exercise 6 (Azizov).

8

Let A be a bounded uniformly J-dissipative operator with VA =.1y. Prove that the that J'+ are uniformly operator A has a single pair of invariant subspaces 9 E definite and Im a(A 122+) > 0, Im a(A I Y-) < 0 ([VI] ). Hint: Use Theorems 1.13 and 2.1.

9

Let A be a maximal J-dissipative operator, let A E (L), 1/A = ., and let A12 be an A22-completely continuous operator. Prove that the operator A has at least one invariant subspace 22+ E .,//+, 9,+ C c/A and Im a(A 19'+) > 0 (cf. Langer [2], M. Krein [5], Azizov and E. Iokhvidov [1]). Hint: Use Theorems 1.13 and 2.8. Let A be a maximal r-dissipative operator (in particular, a r-self-adjoint operator) in

10

H. with c/A = M. Prove that it has invariant subspaces 1'± E .. //± such that Im a(A 12'+) > 0, Im a(A I Y'-) < 0. Moreover, Lin(95,(A) I Im X > 01 C 99+ and C Lin(A'1,(A) I Im X < 01 C -Y'-1 (Pontryagin [1], Azizov [4], [8], M. Krein and Langer [3]. Hint: Use Theorem 1.13 and Corollary 2.9. 11

Under the conditions of Exercise 10 let OA < r (regarding the symbol OA, see 2.Exercise 6 on §2) and let the right-hand (respectively left-hand) boundary ray of the corresponding angle form with the positive (respectively negative) semi-axis an angle

,p, > 0 (respectively ,p2 > 0). Prove that the operator A then has no points of the spectrum within the angles (- roe, ,p,) and (- r + p1, r - a2) and if, moreover, Ker A is definite, then the operator A has a single pair of invariant subspaces 2'± E. // that J'` are definite and J'+ = Lin (21,(A) Ker A n.#,, I Im X > 01, and r- = (Lin (2',, (A `), Ker A fl .? - I Im it > 0)) `11 (Azizov [8] ). 1

3 Invariant Semi-definite Subspaces

176

Fixed points of linear-fractional transformations and invariant subspaces §3

1

Let Y = W+ O+ .e- be a J-space and V = 11

V;; 11

;- 1

be a J-bi-non-

contractive operator. It follows from 2.Theorem 4.17 that on the operator ball + _ ,X+( +, -) (see 1.Proposition 8.19) the Krein-Shamul'yan linear-

fractional transformation Fv: K - Fv(K) _ (V21 + V22K)(V11 + V12K) ',

(3.1)

or (in equivalent form) Fv (K) V11 + F+V (K) V12K - V21 - V22K = 0

(3.2)

is properly defined. 3.1 Let K be the angular operator of the subspace Y+ E 11+. Then Fv (K) is the angular operator of the subspace V P+ (E_lf+ ), and therefore the function

Fv maps the ball X+ into If x = x+ + Kx+, then

.yf+.

Vx= V11x+ + V21x+ + V12Kx+ + V22Kx+ = (Vii + V12K)x+ + (V21 + V22K)x+ = y+ + (V21 + V22K)(V + V12K)-'y+,

where y+ denotes the vector (V11 + V12K)x+. This proposition and the writing of the function F+V in the implicit form

(3.2) enables us to introduce the concept of a generalized linear fractional

transformation Fv defined by a bounded J-non-contractive operator V (V v = A e) as a mapping of elements of the ball Jy'+ into the set of subsets of this ball:

Fv(K)= (L' E.yl+I L'V11+L'V12K- V21 - V22K=0).

(3.3)

The following proposition is proved in the same way as 3.1 was: 3.2 &+

Let V be a J-non-contractive operator, let K be the angular operator of E. if+, and let L be the angular operator of the subspace V5F+. Then

F+V (K) = X+ (L), where .Yl+ (L) is defined by the formula in 1.8.9.

Definition 3.3: A subset J ( C .YC+ is said to be invariant relative to the generalized linear fractional transformation Flt if F+V (K) fl X ;4 0 for any K E N. Moreover, the restriction F+V I W is understood to mean the mapping Fv I J': K (E.X) -+ F+V (K) fl x. In particular, if J consists of a single operator KO, then KO is called a fixed point of the transformation Fv . From Proposition 3.2 it follows immediately that

3.4 A subspace 2' E If+ with the angular operator Ko is invariant relative to a J-non-contractive operator if and only if Ko E F+V (Ko), i.e., Ko is a fixed

point of Fv .

§3 Fixed points of linear fractional transformations

177

We note that this proposition coincides with Lemma 2.2, if in the latter we put T = V, a J-non-contractive operator. Proposition 3.4 shows another way of seeking the solution of the problem of invariant subspaces. This way consists in investigating when the function Fv has a fixed point. In order to apply this idea we need some topological concepts and results; we introduce the latter without proof. 2

Definition 3.5: Let E be a Hausdorff linear topological space, and Jt' a subset of it. A mapping F which carries points K E X into now-empty convex subsets

F(K) C E is said to be closed if the fact that generalized sequences (K6) and ( Fa) (F6 E F(Kb )) converge to KO and Fo respectively implies that Fo E F(Ko).

Theorem 3.6: (Glicksberg [IX] ). Let iC be a non-empty bi-compact convex subset in a locally convex Hausdorff topological space E, and let F be a

closed mapping of points K E J into non-void convex subsets F(K) C X. Then the function F has at least one fixed point in J', i.e., there is a point Ko EX such that Ko E F(Ko).

It is easy to see that under the conditions of Theorem 3.6 the following proposition holds: 3.7 The set of fixed points of the mapping F is closed In our case the role of E will be played by the space of linear continuous operators acting from one Hilbert space into another, and the role of ..W by the ball yl + of this space or its closed convex subsets. The topology is the weak operator topology. We now pass on to the key result of this section.

Theorem 3.8: Let ,' =+ O+ Y- be a J-space and V = I I Vii I I i;=, a J-non-contractive operator with V12 .99-; let Fv be a generalized linear-

fractional transformation generated by the operator V according to the formula (3.3), and let .X ( C X + ) be a non-empty convex subset, closed in the weak operator topology, which is invariant relative to F+V. Then the mapping

Fv I JY has at least one fixed point. The set of fixed points of the function Fit X is closed in the same topology. We verify that we are under the conditions of Theorem 3.6. Let KE X. Then it follows from a comparison of Proposition 3.2 with 1.Theorem 8.23 that the non-empty set Fv (K) is bi-compact and convex, and the same is true of the non-empty set Fv (K) fl X. Thus the function Fv I .X carries points from it' into its convex non-empty subsets. It remains to verify that Fv I it is a closed mapping. Let (Kb) be a generalized sequence of elements from .yl, and let F6 E Fv (K6) fl it' and K6 - Ko, F6 - Fo. From the Definition (3.3) we then have F6 V1 i +. F6 V12K6 - V21 - V22K6 = 0.

3 Invariant Semi-definite Subspaces

178

2.7 F6 V - Fo V11, V22K6 - V22Ko, by Lemma and F6 V12Ks - Fo V12Ko, we have Fo E Fv (Ko). Since i' is closed, Fo E -W, i.e.,

Since

Fo E Fv (Ko) n W. It now follows from Theorem 3.6 that Fv I i' has at least one fixed point, and from Proposition 3.7 that the set of all such points is closed.

3

As a corollary from Theorem 3.8 we obtain

Theorem 3.9: A J-non-contractive operator V (EA_) has the property

J-bi-non-contractive operator V (EA+) has the property

+; a

_; a J-unitary

operator U (EA+ n A _) has the property 4? Ill.

Let V be a J-non-contractive operator and V E A_ . Without loss of generality, by virtue of Remark 2.4, we assume that V12 E .9'm. If a nonnegative subspace 2'o with the angular operator Ko is completely invariant relative to V, then it follows from Proposition 3.2 that the weakly closed convex set V, (KO) is invariant (see 1.Theorem 8.23) relative to the generalized

linear-fractional transformation Fv. Therefore by Theorem 3.8 the function Fv I .W+ (Ko) has at least one fixed point Ko which, by Proposition 3.2, will be the angular operator of a maximal non-negative invariant subspace !'o of the operator V, and Yo E .SPo.

Now let V be a J-bi-non-contractive operator, VE A+. By Remark 2.4 we can suppose that V21 E Y., and therefore (V`)12 = - Vzl E Y- Consequently we conclude from Theorem 3.8 that the function Fv° has a fixed point in any convex weakly closed subset from X+ which is invariant relative to Fv'. In particular, if 3 o C Velo C Mo, and Qo is the angular operator of the subspace Mo, then it follows from the invariance of 9101) relative to V` (see 2. Proposition 1.11) and from Proposition 3.2 that the weakly closed convex subset X-* (Qo) (see 1.Theorem 8.23) is invariant relative to Ft'. Let 20 = (x+ + Kox+ I x+ E+ ), where KO is a fixed point of the function Fv , I X *(Qo). Then 20 is invariant relative to V` and is J-orthogonal to 91o

(see 1.Proposition 8.22). Therefore x[01] (Elf-) is a subspace invariant relative to V and containing 91o. The last assertion about a J-unitary operator U is proved similarly. Namely, because of Remark 2.4 we can suppose that U12 E Y,. If (2'+, Y_) is a dual pair with U T+ = -T+, and if K+ are the angular operators of the subspaces 91± respectively, then, as a convex weakly closed subset X in .y1 + which is invariant relative to Fu, we consider the non-empty subset .W = J'+ (K+) n .X_* (K_ )

(see 1.Theorem 8.23). By Theorem 3.8 Fu I.X has a fixed point K+, and therefore 9+ = (x+ + K+ x+ I x+ E ,Y+) is a subspace which is invariant relative to U, contains Y+, and is J-orthogonal to Y_. Therefore (9+,. '+1]) is a maximal dual pair which is invariant relative to U and contains (Y+, Y_).

§3 Fixed points of linear fractional transformations

179

Corollary 3.10: Let V be a 7r-non-contractive or a a-bi-non-contractive operator. Then it has the property If V is a a-unitary operator, then it has the property c Ill. Since V12 and V21 are finite-dimensional operators, VE A+ fl A_, and it only remains to apply Theorem 3.9.

Corollary 3.11: Let 11, =11+ 0 II_ be a Pontryagin space and U be a 7r-unitary operator. Then U is stable if and only if all its eigen-subspaces Ker(U- XI) are non-degenerate.

Z Let U be a stable operator. In accordance with 2.Corollary 5.20 we can without loss of generality suppose that U11± = I l+, and therefore

Ker(U- XI) = (Ker(U - xI) fl II+) [ (D] (Ker(U- xI) fl Ii-). This equality implies that Ker(U - XI) are non-degenerate.

Conversely, suppose that all the Ker(U - XI) are non-degenerate. Then X E aa(U) implies I X I = 1 (see Exercise 5 on 2.§6). Therefore (see Exercise 22

on 2.§5) Ker(U- XI) [1] Ker(U -µI) when X pe µ, and so there are precisely p (0 < p < x) different eigenvalues X1, X2, .. X, of the operator U to which correspond non-negative eigenvectors. Since the Ker(U- X,I) (i = 1, 2, . . ., p) are non-degenerate, it follows that

,

Ilk = Ker(U- X,I) [+]Ker(U- X2I) [+]

[+]Ker(U- X I) [+].-I

where

.il'= [Ker(U- X1I) [+]Ker(U- X21) [+] ... [+]Ker(U- X I)J and U,4' C 'U in accordance with 2. Proposition 1.11, and by construction U I A has no non-negative eigenvectors. By Corollary 3.10 /V is a negative subspace and therefore U I N is a unitary operator relative to the scalar product - [x, y] 1,/V. Consequently u is a unitary operator relative to the scalar product (which is equivalent to the original one)

(x,Y)1 = L (xi,y;)1=1

where

X= Z; x,+x.,, y= Z; y,+ y.,, i=1

x,,y,EKer(U-X,I) for i= 1,2,...,p,

,_1

and

x,, Y., E./l. From this it follows that the operator U is stable. Theorem 3.9 enables us to make Theorem 2.8 more precise for the case of a J-unitary operator.

3 Invariant Semi-definite Subspaces

180

Theorem 3.12: Let U (EA+ U A_) be a J-unitary operator, let A be its non-unitary spectrum, A = A, U A2, A, fl A2 = 0, and let Az-' (X-' I X E A2) = Ai. Then the operator U has invariant subspaces J e± E such that the non-unitory spectra of U 12+ and U 19_ coincide with A, and A2 respectively. Moreover, A C p(U), and if X E A, (respectively, X E A2), then

the root subspace .If (UII+) (respectively, .x(UI If_)) coincides with -SPa(U). El

A C p(U) by virtue of Remark 2.4 and 2.Corollary 5.13. Let

If+=CLin(.)JU)IXEA1),

2'_=CLin(.If,,(U)IXEA2).

Since Al- ' f1Ai = 0 by hypothesis, .± C 1° by virtue of Exercise 7 on 2.§6. By

construction Y+ are completely invariant subspaces of the operator U and as(UIY+)=F1,(U)( x(UI2_)=Yx(U))when XEAl (respectively, XEA2). By Theorem 3.9 there are subspaces -ie7± E J(± which are invariant relative to

U, which contain .± respectively. Carrying out an argument similar to that used in proving Theorem _2.8 we realize that the non-unitary spectra of the operators U 12+ and U I _ consist of normal eigenvalues. By construction A, C a(U I I+) and A2 C a(U 12_ ), and the corresponding root lineals satisfy the requirements of the theorem. It remains only to notice that the skewconnectedness of .,,(U) and 2'x - ,(U) (see 2.Corollary 3.12) implies that

a(UI.+)nA2=0anda(UI2_)f1A,=0. In this paragraph we investigate the question of the number of invariant subspaces possessed by J-bi-non-contractive operators which have at least one 4

invariant maximal uniformly positive subspace (from not on the sets of uniformly definite subspaces from . C and ./l(- will be denoted by t!o and ..llo respectively). But first we prove the following Lemma 3.13: Let .e = M+ O+ operator of the subspace .ii'- E assertion holds: if the subspace

.

- be a J-space and let Q be the angular Then W = .,Y+ + /V- and the following

Y_(x++Kx+x+Ee+,K: Xe+

-W-, IIKII<°°)

and if 1 E p(QK), then

2=(y++Fy+Iy+E. +, F.

+- 'l-, II FII <1

and - 1 E p(QP- F), and

F= (P + Q)K(I+ - QK) Conversely, if

?=(y++Fy+Iy+E.,Y+,

IIFII
§3 Fixed points of linear fractional transformations

181

and - l E p (QP- F), then

Y= (x+ + Kx+ I x+ E .,Y+ , K: e+ -. ,W- ,

11 K11 < oo

and

1 E p(QK),

and

K = P- F(I+ + QP- F)-'. El

The equality

.

= . + + { "-

(3.4)

is obtained from 1.Corollary 8.16. Let

q'= (x++Kx+ x+E.W+, K:

IjKjj
Then, if x = x+ + Kx+ E Y, we have

x = x+ - QKx+ + Kx+ + QKx+ = (I+ - QK)x+ + (P- + Q)Kx+

= y+ + (P + Q)K(I+ - QK) y+ = y+ + Fy+, where

y + = (I+ - QK)x+

and F= (P + Q)K(I+ - QK)-'.

Therefore

Y _ ly+ + Fy+ I y+ E de+, F .

- .N-, 1 1 F 1 1 < oo J.

The assertion that -1 E p (QP- F) follows from the equality

QP F= QP(P +Q)K(I+-QK)-'=QK(I+-QK) =-I++(I+-QK)-i The other assertions of the lemma are proved similarly. Theorem 3.14: Let V be a J-bi-non-contractive operator, and let SP+ (E,41+ ) be an invariant subspace of V Then the operator V has an invariant subspace {.E -11-. If, moreover, V has an invariant subspace (9?+ # ).ii 1 E /tf+ such "that t'l + A= W, then this operator has not less than a continuum of

invariant subspaces in each of the sets ./tfo and /tf+\,-tfo .

JLet K+ be the angular operator of the subspace -V+. It follows from 1.Lemma 8.4 that II K+ II < 1, and by Lemma 2.2 Gv (K+) = 0. Therefore VE A+ and so, by Theorem 2.8, V has an invariant subspace A- E ..Jf-.

We now use results from 1.§7.6 and we shall suppose, without loss of generality, that 2+ _ ,Y+. Let Q be the angular operator of the subspace A"

and K, the angular operator of a subspace ./Vi (*.,Y+). By virtue of 1.Theorem 8.15 assertion c) 1 E p(QK,) and so it follows from Lemma 3.13 that.4"1+

= (y+ + Fly+J, where y+ E.W+ and F, = (P + Q)K,(I' - QK,)-'.

3 Invariant Semi-definite Subspaces

182

Moreover F, ;e 0. Since

+ and

V- are invariant relative to V, and

+ Fi: the subspaces . 'l G_' = (y+ + a F, y+ I y+ E W+) will also be invariant for all a E C. We note that the G+ are not necessarily semi-definite. {

But, for a sufficiently small in modulus we have - 1 E p(aQP- Ft) and therefore when, for example, I a I < z II F, II ', it follows by Lemma 3.13 that ,/I-Q+ =

(x++Kax+Ix+.

+, K.,=aP-F,(I++aQP-F,)-'} (3.5)

and since Ia1IIF1I1 <1 IIK.II< 1-a1IIF111

it follows that all such iV are uniformly positive. Moreover the power of the sets of them is not less than that of the continuum. Without loss of generality we can now suppose that J V E ,? , i.e., II K, II < 1 ;the equality IV; + V- = M is obtained from 1.Corollary 8.16). We consider the set of those a E C for which - 1 E p(aQP- F). This is an open set in each connected component of which the function a - K. is holomorphic. Let Z be the connected component of this set which contains the point of = 1 (see Lemma 3.13). We prove that there is a point ao E -Z such that II K.. II > 1. First of all we notice that if - P6 C, then Z has a finite boundary point CL'.

Then, if a E Z and a - a', we conclude from the form of the operator K. oo as n oo, and the existence of ao has been proved. But if ,?. = C and if, contrary to assumption; I K, II < 1 (a E C), then by Liouville's theorem K. = constant. Therefore QKq = aQP- F, (I+ + aQP- Ft)-' =

(see (3.5)) that II K,, II

I+ - (I+ + aQP- F1)-' = const., which implies the equality QP- F, = 0, and therefore K. = aP- F, 0 const. (because F, ;4 0)-we have obtained a contradiction. We now consider continuous functions

(a = )ff: [0, 1]

(0 < T < 1)

with

f,(0) = 1 and f,(1) = ao

for all rE [0, 1] and

A(t2) when T1 ;4 r2 and t1, t2E (0, 1).

Since all the functions fl(t) are continuous, the functions II Kf(t) II will also be

continuous with respect to t for each r, and moreover IIKf(o)II=IIK,II < 1 and

1,

and therefore there are t,E (0, 1) such that II KJ,(,,) II = 1. By hypothesis ;e-

when Ti ;4 T2. Therefore the operator V has not less than a

continuum of different invariant subspaces ./l "f (,_) E .,//+\. //o with the angular operator K,(,_).

§3 Fixed points of linear fractional transformations

183

In conclusion we shall show how to obtain from results about invariant subspaces for J-non-contractive operators similar results for operators acting in G-spaces. In 1§6.6 a construction was given for embedding a G-space in a J-space, in which, in particular, a G(')-space was embedded in IIX. We shall call such an embedding canonic, and we shall investigate what happens to certain properties of operators acting in the corresponding spaces. Later we shall use the following general theorem which we introduce here without proof. 5

Theorem 3.15 (M. Krein [2]): Suppose a Banach space R is densely and continuously embedded in a Hilbert space W. If a linear operator A: V - V with VA = 4 is continuous (respectively, completely continuous) in 4 and is also symmetric as an operator in Ye, then it is continuous (respectively, completely continuous) in ' and it can be extended by continuity on to the whole of .

into a self-adjoint operator A: ' e.

As a direct consequence of this theorem we obtain the following proposition which will be important later: Lemma 3.16:

Let the G-space.,Y = Ae® +- with G.± C W', and let .

be canonically embedded in a J space 7e _ + O+ - with ± canonically embedded in R±. If A with 9A = M is a G-selfadjoint continuous (respec-

tively, completely continuous) operator, then it extends uniquely into a continuous (respectively, completely continuous) J-selfadjoint operator A

R.

Let J (= I G I -G) be the unitary part of the polar decomposition of the operator G; in the given case J coincides with the difference of the ortho-

projectors P+ and P- on to+ and- respectively: J = P+ - P-. Then JA is a I G I -selfadjoint operator, and . - by construction (see 1.Proposition 6.14) In accordance with is the completion of i( relative to the norm (I G I x, x)1,12.

Theorem 3.15 the operator J extends into an operator J = P+ - P- (P`- are the ortho-projectors in .W on to ±), and JA extends into an operator JA selfadjoint in W which will be simultaneously with A continuous or completely continuous. Therefore A = Y JA is the required operator. Corollary 3.17: I f under the conditions o f Lemma 3.16 A = I I AU II' i=1 is a continuous G-selfadjoint operator and A 12, A21 E 9-, then A = II Au II ?i=1 where the A;i are the extensions of the operators A, (i, j = 1, 2) and moreover A12,A21 E .q',,, Since

A = Al + A2, where Al = II

All

A22 I

,

A2 =

0

A 12

A21

0

i

3 Invariant Semi-definite Subspaces

184

A, and A2 are G-selfadjoint operators, and A2 E Y., we have only to use Lemma 3.16.

Remark 3.18:

Suppose W + W+ O+ M- is a G-space, G.W± C e±, G± = G j .Y±, and 0 E p(G_ ). Then from A =Ac, 9A = .Ye and A12 E .y'0 it follows that A21 E 9'm. This follows from the equality A21 = G_'A12G+ which holds in this case.

If V`, together with V, is a continuous operator and then V and V` extend into continuous operators V and Vc in .,Y, and P = V`. Remark 3.19: v = 9 v, =

,

This follows immediately from Lemma 3.16 if we use the equality V= 2'(V+ V c) + i [ (V - V')12i] and the fact that the operators '(V+ V c) and 2

112i(V- V`) are G-selfadjoint. Theorem 3.20: Suppose . = W+ ( Ae- is a G-space, GW± C M±, G+ = G I .e±, and 0 E p(G_ ). If V is a G-bi-non-contractive operator (i.e., Vand

V` are continuous G-non-contractive operators, 9v = 9)V'= .'), V12 E 91., 9'+ C .?±, V9 -?+ = Y+, then there is an 9+ E ill+ (.e) containing q+ and such that V9+ C L+. Let M = . + O+ . - be canonically embedded in ie- = Se- + O+ .ie- -. It follows from 0 E p(G_) that.- = .e-. The operator V satisfies the conditions in Remark 3.19 and it therefore extends into a continuous operator V which is J-bi-non-contractive. Since V12 E 91m and 0 E p (G_ ), so V12 E Y.. Therefore, by Theorem 3.9, the operator V has the property -6 +. The condition V. + = Y+ implies the complete invariance of the closure of 2'+ in ' relative to the operator V. Consequently there is an ! = [ x+ = Kx+ I x+ E .3W+ } (E. #+ (.e)) containing

'+, and V2' =

It is easy to see that 2+ =.t fl '(E. lf+ (M)) will be the

required subspace.

Corollary 3.21: If Atis a Gt"1-space and if V is a G(')-bi-non-contractive operator, then any of its completely invariant non positive subspaces admits extension into a maximal non-positive invariant subspace.

We are in the conditions of Theorem 3.20 to within a change-over to the anti-space and taking account of the fact that V21 is a finite-dimensional operator. So V21EYm.

Exercises and problems I

Let V be a J-non-contractive operator. We define a generalized linear-fractional transformation F v' on a non-empty subset .7'(C.W + ): t V (.i) = Ul F V (K) I K E .J

.

§4 Invariant subspaces of a family of operators Prove that if

y3 = ( V) is

185

a semi-group of J-non-contractive operators, then

F'',j= (Fv I V E .Y31 is also a semi-group in relation to superposition of transformations and that F+vo o Fv2 = Fv, v,; if .13 is a group, then F. is a group, and 2

(Fv)-' = Fir Prove that if U is a J-unitary operator, then

11 K II < 1

Fu (K) < 1 and

11 Ku = I « I Fu(K) II = 1 (M. Krein and Shmul'yan [4], Potapov [1]). Hint: Use Proposition 3.1. 3

Let V be a J-non-contractive operator, U a J-unitary operator, and W = U-' VU. An operator Ko E k is a fixed point of the transformation F v+ if and only if Fij-(Ko) is a fixed point of the transformation F+w.

4

Prove that the function Fv generated by a J-non-contractive operator V E A_ has at least one fixed point in .W (I. Iokhvidov [10]). Hint: Use Proposition 3.2 and Theorem 3.9.

5

Suppose V is a J-bi-non-contractive operator, and F%' has a fixed point Ko with 11 Ko II < 1. Then the function Fv has either a single fixed point in .Yr+, or it has not less than a continuum of such points, and moreover not less than a continuum on the

boundary of the ball i+ (Azizov; cf. Khabkevich [5] ). Hint: Use the result of Problem 3 and suppose Ko = 0. Use Theorem 3.14. 6

Suppose TE A_ is a J-bi-non-contractive operator, (91+, '_) is a dual pair, Y'+ is completely invariant relative to T, and '_ is invariant relative to T`. Then there is an extension of this pair into a maximal dual pair with preservation of the properties mentioned (Azizov). Hint: Suppose T12 E .y'm and use 1. Corollary 8.24 and Theorem 3.8.

7

Prove that if U is a J-unitary operator, then the function Fu is continuous in the weak operator topology if and only if U12 E rm (Helton [3] ).

8

If VE A_ is a J-semi-unitary J-bi-non-contractive operator, then it has the property c (Azizov). Hint: Use Proposition 2.5 and Theorem 3.9.

9

Suppose V is a J-non-contractive operator, Y+ (C .?+) is its completely invariant

subspace with angular operator K+, and def Y+ < oo. Then V is a J-bi-noncontractive operator and there is a subspace Jv+ E ..tl+ which is completely invariant relative to it and which contains Y+ (Azizov). Hint: Verify that Fv is a closed transformation of the convex non-empty bicompact subset . V+ (Kr ), and use Theorem 3.6. 10

Suppose A E (L) is a J-selfadjoint operator, A is its non-real spectrum, A = A, U A2, A, = A2*, A, fl A2 = 0. Prove that if A is an A22-completely continuous operator, then the operator A has a maximal non-negative invariant subspace Y'+ C 9A such

that the non-real spectrum of the operator A 12+ coincides with A, (or with A2) (Langer [3], M. Krein [5]). Hint: Use Theorems 1.13 and 3.12.

§4 1

Invariant subspaces of a family of operators We have already encountered invariant subspaces of groups of J-unitary

operators in 2.0.3. From 2.Theorem 5.18 it follows, in particular, that if W = ( U) is a bounded amenable group of J-unitary operators, then without

3 Invariant Semi-definite Subspaces

186

loss of generality these operators can also be supposed to be unitary. So below,

instead of the condition of amenability and boundedness, we shall immediately impose on a group of J-unitary operators the condition that the operators entering into the group are unitary, i.e., U.W± _ .±. The following theorem makes 2.Theorem 5.18 more precise. Theorem 4.1: If W = ( U) is a group of operators which are simultaneously J-unitary and unitary, then it has the property 4) [1] .

Let ('+, &_) be a dual pair invariant relative to the group 40/, and let be its extension into a maximal invariant dual pair. Since the properties of being J-unitary and simultaneously unitary are -invariant (see Exercise 2 on § 1), we can suppose without loss of generality that (2+, 0-) is a

definite pair (see Exercise 6 on § 1). Since W'+ = k+, and all U E %?l are J-unitary, it follows that 6?l9+l1 = 2+1] . The duality of the pair (2'+, 2_ ) implies the inclusion 2'_ C 21+'1. Since (see 1.(10.1)) 2+11 = 9+ [(@ j ,I', (in the

present case the 4No which appears in 1.(10.1) is equal to [0)), and since all U E ?l are simultaneously J-unitary and unitary which implies Q/!2. + = 9 +, it follows that UN1 =,ii",. By virtue of 1.Theorem 10.2, in order to establish the inclusion 9- E It-, it is now sufficient to establish that TO_ is the non-positive subspace which is maximal in 2+1]. Let P, be the orthoprojector from .+11 on to .4i",. Since U22- = 9_ and U,,11'1 =,4 , we have UP,._ = P, L'_, U[(P,k_ )1 fl <4 i] = (P,9-) -L fl .ii",. The subspace and therefore . Consequently, (P,1- )1 fl i , (C JP+11) is negative and J-orthogonal to

T = Lin (9_, (P,'_ )1 fl +j) is the maximal negative subspace from 9+11 which is invariant relative to all UE 41. The maximality of_ as an invariant negative subspace from 91+'1 relative to the group Q?z implies the coincidence of _ with 21. That 2'+ E t?+ is verified similarly.

In this paragraph we investigate the question of invariant subspaces for groups of normal J-unitary operators. But first we prove some auxiliary 2

propositions.

Lemma 4.2: Let . ' be a J-space, U be a J-unitary and simultaneously positive operator, and let Ex be its spectral function. Then

a) 91, = C Lin (ExN' I X < 1)and2'2=CLin{(I-E,). Iµ> 1)areneutral mutually orthogonal subspaces; b) .0 = Ker(U - I) [ G) ] (2', G) Y2); c) J(SL', (1 2'2) = Y, O+

'2, J Ker(U - I) = Ker(U - I);

d) every definite invariant subspace of the operator U lies in Ker(U - I). a) Since Ea.

C

when X > 2', and 1 - E ).Yf C (I - E,,).*" when µ > µ,

§4 Invariant subspaces of a family of operators

187

to prove the neutrality of 2, and 2'2 it suffices to verify the neutrality of the subspace Ea.' when 21 < 1 and the subspace (I - E,,).' when µ > 1 respectively. From the spectral theory of self-adjoint operators it follows that or (U I E),.') C [ Xmin, X1, where Xmi,, = inf ((Ux, x) 11 x 11 = 1); moreover 0 E p (U)

because U is a J-unitary positive operator, and therefore Xmin > 0. The subspace E,,. is also invariant relative to the operator U` (= U-' ), and Since X<1 by a (U` I Ex. ) C [ 1 / X, I Xmin] hypothesis, so [Xmin, x] n [1/X, I Xmin] = 0. The neutrality of E,,.1f now follows from 2.Theorem 1.13. The neutrality of I = E,). when µ > 1 is verified similarly. The orthogonality of 2, and 22 follows from the orthogonality property of the

spectral function: Ex (I - E,) = 0 when X < and µ > 1. b) The orthogonality of Ker(U - I) to 2, and .?2 follows from the orthogonality of Ker(U - I) to EX.Y( when X < 1 and to (I - E,). when µ > 1.

We now verify that Ker(U - I) [1] 2, and Ker(U - I) [l] 2'2. Because of the continuity of the J-metric it suffices to verify that Ker(U - I) [1] Ea.' when X < 1 and that Ker(U - I) [1] (I - E,).W when µ > 1 respectively, but this in turn follows directly from 2.Theorem 1.13 when we take into account that a(U` I Ker(U - fl) = (1). We consider the subspace .' = Ker(U- I) [ S] (2, Q+ 22) which is invariant relative to U; we shall prove that W' = e. For otherwise we would have . _ .e' G) W ' 1 where ' 1 (7 0) is a subspace invariant relative to U, and U' = U I ,W" 1 is a positive operator. Let Ex' be its spectral function. It is easy to see that E,,.' 1 C E for all v E R. Therefore, since Ex.Y' 1 is orthogonal to E),.t (C 9?1 C Jr") when X < 1 and (I - E, ). Y' 1 is orthogonal to (I - E,, ).w' (C 22 C . ') when µ > 1, it follows that Ex ' 1 = (01 for X < 1

and (I- E,), ,Y' = (0) for µ > 1, and so a(V') _ (1). Hence we conclude that V' = I Ye ' 1 , which contradicts the orthogonality of W ' 1 to Ker(U - 1).

Consequently Ae' =. c) Taking into account 1.Formula (7.1), this follows from b). d) Let 2 be an invariant subspace of the operator U. It is well-known (see, e.g., [XXII] and cf. 4.Remark 1.8) that then ExS C 2. Since Eat is a neutral

subspace when X < 1, the definiteness of 2 implies Eat = (0). Similarly (I - E,)9? = (01 when µ > 1. Therefore X1(2, O+ 22), i.e., 2 C Ker(U - I). Corollary 4.3: Let the operator U satisfy the conditions of Lemma 4.2, and W defined on W let emu be the algebra of all continuous operators A: which commute with U. Then -Bu has a common non-trivial neutral invariant

subspace if and only if U;4 I. Each such subspace is J-orthogonal to Ker(U -

I).

If U = I, then Wu coincides with the algebra of all continuous operators

A: . -+ ., which, as is easy to see, has non-trivial (and including also neutral) invariant subspaces. But if U ;e I, then, for example the 2, and .?2 appearing in Lemma 4.2 are

188

3 Invariant Semi-definite Subspaces

neutral and invariant relative to -j6u, since each of the operators of this algebra commutes with Ex for all X E IR (see, e.g., [XXII] and cf 4Theoreml.5). Let 2 be any non-trivial neutral subspace invariant relative to .tiBu, and let P

be the J-orthogonal projector from ' on to Ker(U - I). Then, as is easily verified. P2' is an invariant subspace of the algebra u I Ker(U - I), which coincides with the algebra of all continuous operators acting in Ker(U- I).

Therefore either P2' = (0) or P2 = Ker(U - I). Since (P` = ) P E X u, so PY C 9 and therefore P-T is a neutral subspace. It follows from assertion b) in

Lemma 4.2 that Ker(U- I) is a projectionally complete subspace, and therefore PL = (0), i.e., 2'[±] Ker(U- I). The following lemma is of a general character and seems to be well-known.

Lemma 4.4: If Y = ( V) is a group consisting of normal operators and containing the conjugate V* whenever it contains V, then VU*U = U*UV for any U, V E Yl.

By hypothesis the operator W= V*VU*UE I' and therefore it is normal. Since W is a (U*U)-selfadjoint operator and the scalar product (U*U , ) is equivalent (in the sense of equivalence of the corresponding norms) to the original one, so a(W) C FR, which, taking the normality of W into account, is equivalent to its selfadjointness: W= W*, i.e., V*VU*U= U*UV*V. As is well-known (see. e.g., [XXII]) it follows from this that (V*V)12'2U*U= Let V= be the polar representation of the normal operator V; here S is a unitary operator commuting with U*U(V*V)1,12.

S(V*V)1,12

(V*V) 1/2. Since S(U*U)25-1 V*V= (V*V) 1/25(U*U)25- 1 (V* V) 112 = (VU*U)(U*UV)

= (U*UV*)(VU*U) = (U*U)(V*V)(U*U) = (U*U)2V*V,

so S(U*U)2 = (U*U)2S. We conclude, as above, that SU*U= U*US. Consequently VU*U= U*UV.

We turn now to the formulation and proof of the main result of this paragraph. Theorem 4.5: Let ,Y be a J-space and 4/ = { U) be a group consisting of

normal J-unitary operators and containing the operator U* whenever it contains U. Then =V! has the property

W.

Let (2'+, 2'-) be an invariant dual pair of the group ail, and let (2'+, ) be its extension into a maximal invariant dual pair. We use Exercise 6 on § 1 and we shall suppose that (2P+, 2'-) is a definite dual pair. By assertion d) of Lemma 4.2 we have 2'± C n{Ker(U*U- I) UE -V!). Moreover, the maximality of J+ implies the equality U*U = I for all U E W. For, if we had Uo*Uo ;d I for some Uo E -V/, then it would follow from Lemma 4.4 and

§4 Invariant subspaces of a family of operators

189

Corollary 4.3 that the group 4! has a common non-trivial neutral subspace

J-orthogonal to Ker(Uo Uo - I) and all the more to k+-so we have a contradiction. Thus U*U = I for all U E 4l, i.e., 4! is a group consisting of operators simultaneously J-unitary and unitary. By Theorem 4.1 it has the property 4) 1-1, and therefore 9± E -11:2:.

Corollary 4.6: If -V/ = ( U) is a commutative group consisting of normal J-unitary operators, then the minimal group containing 4l and W* has the property I']. The set 4!* = (U* I U E =R!) is also a commutative group consisting of

normal J-unitary operators. Moreover, if U, V E 4l, then UV= VU, and therefore by a well-known theorem of Fuglede U*V = VU*, i.e., the elements of the groups 4! and 41 commute with one another. Consequently, the group

FV generated by the union of 4l and 4l* is commutative and consists of normal J-unitary operators. Moreover, if UE 41, then U*E Ql. It only remains to use Theorem 4.5. The operator ball .yl+, as was shown in I. Proposition 8.20, is bicompact

3

in the weak operator topology, and therefore (see 1.Proposition 8.21) the centralized system of its closed subsets has a non-empty intersection-on this is based the proof of the following key proposition. Theorem 4.7:

Let Y' = ( V) be a family of J-bi-non-contractive operators, let F; = (Fv I V E Y ) be the corresponding linear fractional transformations of the ball JY (see (3.1)), and let .Ylb be the closed set of fixed points of the transformation F. If for each finite set (Vi) C 'F we have ni .X<, ;d 0, then n(.XirI VE Y' E Y') d 0. Corollary 4.8: Let Y = ( V) be a family of J-bi-non-contractive operators V with Vie E 9. (respectively, V21 E Y.). If each finite subset of Y has the then the whole family Y has the property D+ or 4 + (respectively, (D_ same property. If Y is a family of J-unitary operators with V1 2 E .gym (or, what is equivalent, V2i E

then similar assertions hold also regarding the properties 4'[11

and $110.

Let V be a J-bi-non-contractive operator, and let JYv be the set of fixed

points of the function F. By Theorem 3.8, when V12 E Y., this set J' is closed in the weak operator topology. By Proposition 3.4 each finite subset Y ' from Y has the property 4'+ if and only if n(.X v I V E Y -) # 0, and Y having

the property 4'+ is equivalent to n(.XvI VE Y') ;d 0. it only remains to use Theorem 4.7. The remaining assertions are proved similarly.

3 Invariant Semi-definite Subspaces

190

Corollary 4.9: If W= (U} is a commutative family of stable J-unitary operators and if U12 E 9. for all U E /!, then W has the property i [1] . Let U1, U2,. . ., U" be an arbitrary finite set of operators from i, and let Wn be the group generated by these operators. From the commutativity of W and the stability of the operators comprising W, we conclude (see II, items 5.17-5.20) that -Wn is a bounded amenable group. Therefore (see §4.1) -W has the property 4)[1]. It remains to apply Corollary 4.8. Under the conditions of Corollary 4.9 the whole family %1l is not necessarily bounded. The following example confirms this.

Example 4.10: Let I1 =1I+ O+ n- be a Pontryagin space, II+ = Lin(e+}, and II_ = C Lin{e-}i , (ei , e;) = Si; (i, j = 1, 2, ...), and let the sequence (;}i of complex numbers be such that Ei 1 tr 2 = 1, E; ;4 0 (i = 1, 2, ...). We consider the subspaces I

n

II In) = C Lin e+ +

Ete;, i=1

l

en-+2, ... }.

These too are Pontryagin spaces with x = 1, IIf") of"+'>, III")[-L] are negative subspaces, and II1 = of") [ + ] II(") [1] (n = 1, 2, ...). We define operators Un:

Unx = x when x E

of")'

Uny = - y when y E

It is easy to see that these are a-unitary stable operators (because U = I) and UnUn = U,nUn (n, m = 1, 2, ...). It can be seen from the construction of the operators Un that all their non-negative eigenvectors lie in III'), and therefore the non-negative eigenvectors of the family W = (Un) 1' lie in fl (IIf"t}; = Lin(e+ + E 1 ,e; ). Thefore JI/ has a single non-negative invariant subspace Lin (e+Ei E,e; } and it is neutral. But if the family ! were bounded, then the group generated by it would also be bounded. By 2.Theorem 5.18 this group, and therefore also the family Wl, would have to have a positive invariant subspace-a contradiction. 4 We pass on now to the investigation of the question of invariant subspaces for a-non-contractive operators. Theorem 4.11: Let Y = (V } be a commutative family of a-non-contractive operators in H,,. Then it has the property 4?.

In accordance with Corollary 4.8 it suffices to verify the assertion for a family Y consisting of a finite set of operators. We verify that such a Y has the property +. Let Y+ (C e°+) be an invariant subspace of this family, and let 9+ be its maximal invariant non-negative subspace containing M1+. We shall use Exercise 4 on § 1 and we shall suppose that 9+ is a positive subspace.

§4 Invariant subspaces of a family of operators

191

+ [+]+1', where 9+11 is again a Pontryagin space flx, with = x - dim Y+. We shall prove that x, = 0, i.e., that .'+ E ,u+.

Then II =

Suppose x, > 0 and let P be the a-orthogonal projector on to +1J . Then, as is easily verified, the operators V, = PV 12i+] form a commutative family y

, = ( V,), and moreover the V1 are 7r-non-contractive operators in

2P+11 (see

the more general Lemma 5.9 below. Since 9+ is the maximal invariant non-negative subspace of the family Y, it follows, on the one hand, that Y', has no non-negative invariant subspace 2, (B}: for otherwise Lin(9+,2,} would be an extension of 9+ into a non-negative invariant subspace of the family Y' and on the other hand, by Corollary 2.9, each of the operators V, has in Yl+'l a x,-dimensional non-negative invariant subspace. Moreover, if V, xo = Xxo, with (B ;4)xo E JO+, then I X I = 1. For, if I X I > 1, then, by Exercise 5 on 2.§6, 21,(V,) is a non-negative subspace and it is invariant relative to Y-which is impossible. But if we had I X I < 1, then (see Exercise 5 on 2.§6) xo E -, i.e., xo 410, and the isotropic part of 2a(V1) would again be invariant relative to y, which once more is impossible. Therefore, I X I = 1 and Ker( V, - XI) is non-degenerate. Consequently (see Exercise 23 on 2.§4)

the operator V, has no associated vectors corresponding to this X. Let X1, X2, ..., X, be all such points from a,(V,) to which correspond non-negative eigenvectors. It is clear that', = Lin (Ker( V, - X,I)) ° is a certain Pontryagin

space H. From Corollary 2.9 we conclude that x = xl. Moreover, ,)'1 is invariant relative to 'Vi. We consider the family 111 _ V1 01 = ( V I

, ). It

again is commutative and consists of 7r-non-contractive operators one

of which is, as is easily seen, a stable 7r-unitary operator. By carrying out the procedure indicated as many times as there are operators in Y/, we obtain a family 7i' = ( V') consisting of a-unitary operators which are the restrictions

of the original operators on to an invariant subspace IIx, with x' = x,. According to Corollary 4.9 the family '//' has a x1-dimensional non-negative subspace-we have obtained a contradiction. We now verify that the family Y has the property 4_ . Let Je_ be the maximal invariant non-positive subspace of the family, containing the original one. Again by virtue of the result of Exercise 4 on § 1 we can suppose that 1- is a negative subspace. Then 9-I'] is a Pontryagin space fl,', with x' = x and it is

invariant relative to the family Y" = (V` I V E Y } consisting of it-noncontractive operators. In accordance with what we have proved above, Y ` has a x-dimensional non-negative invariant subspace Y+ in *U1; but then ii'L+)

(E,11-) is invariant relative to Y and Y+> D 9-, and this is possible only when Yl+11 coincides with_ .

If ! = (U) is a family of pairwise commutating ir-unitary operators, then it has the property 4fll. Corollary 4.12:

Let (9+,.9'-) be the maximal invariant dual pair of the family W, containing the original one. In accordance with Exercise 6 on §1 we can

3 Invariant Semi-definite Subspaces

192

suppose that (2'+, 2'-) is a definite pair, and therefore [9+ [+19, ] [ll is a Pontryagin space rI,,, invariant relative to 1l with x' = x - dim 2'+ By Theorem 4.11 when x' > 0 the family 4! has in I1x' a x'-dimensional non-negative invariant subspace 2, and a maximal non-positive invariant subspace .'z= P;11 f H. Consequently (Lin(9+,.2,), Lin(Je1_,2'Z)) is an extension of the dual pair (2'+, 2-) preserving invariance relative to ill, and this is possible only when 91, = 2'2 = [0), i.e., k± E J1

.

Let .e be a G(")-space, let ail= (U) be a commutative family of G(')-unitary operators, and let 2'_ be its maximal invariant non-positive subspace. Then 2- E U. Corollary 4.13:

We use the results of §3.5 and canonically embed the G(')-space W in II.,

and we also extend the family 41 by continuity into a commutative family %1l = (U) of 7r-unitary operators. In accordance with Corollary 4.12 Wl has the

property c -, and therefore there is an 2 E - (ILL) which contains 2'_ and is invariant relative to TV. Then (cf. 1.Proposition 8.18) 2'fl E mil- O, 22 fl W' 3 2'-, and 2' fl W' is invariant relative to JIV. Since 2'_ is maximal it

follows that 2'_ = k fl Y. The result obtained enables us to prove a series of propositions about the existence of invariant subspaces; one of them is

Theorem 4.14: Let 4! = (U) be a commutative family of J-unitary operators, let (2+, 2-) bean invariant dual pair of 4l with def 2'+ < oo or def ,'_ < cc, and let (2'+, 2'-) be its extension into a maximal invariant dual pair. Then 2+ E It+.

Without loss of generality we shall suppose that def 2'+ < co, and so def 2_+ < co also. In accordance with Exercise 6 on § 1 we can suppose that

(2'+,2'-) is a definite dual pair. Consequently 9+11 is a G(')-space with x = def i+, the W 191+'1 are G('')-unitary operators, and 2' is their maximal

invariant non-positive subspace. We conclude from Corollary 4.13 that 2'_ E tl- (2'1+-LI), and so (cf. I.Theorem 10.2) 2'_E ll ( ). Therefore (x!11, 2'_) is an invariant dual pair containing (2'+, 9-), and since the latter is maximal we obtain 9+ = E ,tl+

Exercises and problems I

Investigate whether in Theorem 4.5 the condition (U E N!) _ (U* E -Y!) can be omitted.

2

Let Y = (V) be a commutative family consisting of a-non-contractive and ir-bi-nonexpansive operators. Then it has the property 4i (Azizov (6)).

3

Generalize Corollary 4.13 to the case where i+! _ (U) is a commutative family of G`-non-contractive operators and VU = VU = N' (Azizov).

§5 Operators of the classes H and K(H)

193

4

Prove that if -VI = ( Ul is a commutative family of J-bi-non-contractive operators, if 1'+ (respectively, 2'_) is a maximal completely invariant (respectively, maximal invariant) non-negative (respectively, non-positive) subspace of the family -f/ and def

5

Let W = Lin (e, fl, 11 e 11 = 11 f 11 = 1, (e, f) = 0. We introduce into . ' a J-metric by means of the operator J: J(ae + /3f) = /3e + af. Prove that the group generated by the operators J and U: U(ae+Of)= Xae+X-'(if (I a ;4 1) is soluble, consists of

'+ < co (respectively, def 2'_ < oo), then Y+ E fl+ (respectively, Y_ E Lf-) (Azizov).

jr-unitary operators, and has no common non-trivial invariant subspaces (cf. 2.Theorem 5.18) (Azizov). 6

Let d be a commutative algebra of operators acting in a J-space and closed relative to

the operations of conjugation and J-conjugation, i.e., A E d - A * E d, A` E d. Prove that if (Y+, °-) is any maximal dual pair invariant relative to d, then T± E ./ff± (Phillips [3] ). Hint: Use Theorems 1.13 and 4.5.

§5

Operators of the classes H and K(H)

In 1.§5.4 we introduced the concepts of the classes h± with which we shall operate in this section. 1

Definition 5.1: We shall say that a bounded operator T belongs to the class H

(TE H) if it has at least one pair of invariant subspaces 2+ E ..Zf+ and Y_ E -&- and every maximal semi-definite subspace Y± invariant relative to T belongs to MI respectively. From this definition and 2. Proposition 1.11 the implication

TEHa T`EH

(5.1)

follows immediately. We now investigate a number of other properties of operators of the class H.

Theorem 5.2: If an operator T has an invariant subspace of the class -it, fl h+ (respectively, .,L(- fl h-), then T E A+ (respectively, T E A_ ). In particular, T E H= T E A+ n A_ . D

Let K+ be the angular operator of the invariant subspace 99+ E

of the

operator T. If 2+ E h+, then K+ can be expressed in the form of a sum K+ = K1 + K2, where 11 K1 11 < 1, and Kz is a finite-dimensional partially isometric operator. By Lemma 2.2 Gi (K+) = 0, and therefore

GT(Ki )(K+-K2 )T>>+(K+-K2 )T12(K+-K2 )T21-T22(K+-K2 ) _ -K2+T11-K2 T12K+-K+T12K2+-K2+T12K2+ +T22Kz E.V and by Definition 2.3 TE A+. Similarly one proves that if the operator T has an invariant subspace _T- E , ll- fl h-, then TE A_. From Definition 5.1 and what has been proved, it follows that TE H = TEA+ n A-.

3 Invariant Semi-definite Subspaces

194

It follows from this theorem that the propositions proved earlier (see §§2-3) for operators of the class A+ hold also for operators of the class H. In particular, from Theorem 3.9 and Exercise 8 on §3 we obtain

Corollary 5.3:

If T (E H) is a J-bi-non-contractive operator, (respectively, a

J-semi-unitary J-bi-non-contractive operator or, in particular, a J-unitary operator), then T has the property c (respectively, Ill ). Corollary 5.4: Let T (EH) be a J-bi-non-contractive operator. Then each of its completely invariant non-negative (respectively, invariant non-positive) subspaces belonging to the class h+ (respectively, h-). It is sufficient to use Corollary 5.3, the Definition 5.1, and the simple fact that a subspace of a subspace of the class h± belongs to h+-.

Let T be a J-bi-non-contractive operator of the class H. Then

Corollary 5.5:

there is a constant xT < oo such that the dimension of each of the neutral invariant subspaces of the operator T does not exceed xT. Let (2') be the set of all neutral invariant subspaces of the operator T. It follows from Corollary 5.4 that dim 2 < oo for all ' E (i). If we assume that there is no constant XT < oo bounding the dimension of the subspaces Y, then with dim 9?;, oo, among them, there could be found a sequence dim 2;, <, dim We put Yi = Yi,

= Lin 191,,

(n = 2, 3, ...).

Y,;n

is a monotonely non-decreasing sequence of neutral invariant subspaces of the operator T, and dim 2 - oo as n - oo. But then is also a neutral invariant subspace of the operator T, and C dim C oo-we have obtained a contradiction. By construction

Corollary 5.6: Let T (EH) be a J-bi-non-contractive operator, and let Ker(T - XI) =,4'0\ [ + ],4'x+ [ + ] . l "X be the canonical decomposition of the kernel of the operator T- XI into an isotropic component A 'OX, a positive Then .4!"a are uniformly component .4-x+, and a negative component difinite subspaces and min{dim(.4'Q [+].-!"x; ),dim(.'l"0 [+].'!" )) <

From Corollary 5.4 we obtain that dim A"0 < oo, that ./l"a are uniformly definite subspaces, and that any definite subspaces in A "X [ + 1, /l "x are uniformly definite. Therefore by virtue of 1.Theorem 9.11 which .'! "a , dim .4) < oo, min (dim(.'!"0 [+].'!",; ), dim(.' U. [+]A "a )) < oo. min (dim

implies

the

inequality

Let T (XH) be a J-bi-non-contractive operator. If T has no neutral eigenvectors, then it has a single pair E #+ and 99- E -I of Lemma 5.7:

Y+

§5 Operators of the classes H and K(H)

195

invariant subspaces and they are uniformly definite, .' = 2+ -- 2-. Further-

more, if T is a J-semi-unitary or J-unitary operator, then 97 = Y+ [1] and

j'_9+[+]2'

.

Let 2± (E_#) be invariant subspaces of the operator T. By definition 5.1

T+ E h±, and therefore its isotropic part 9o is finite-dimensional and invariant relative to T. Therefore if T has no neutral eigenvectors, then Yo = (B], i.e., the subspaces Y+ are uniformly definite. By Theorem 3.14 and 1.Corollary 8.16 it follows that the 2'± are unique. Therefore if, in addition, T

is a J-semi-unitary operator, then TY+ = 2+ implies T`2' _ 9?+, and so TY+ [1] C 2+ I`]. Since the pair 2+ and 27 is unique we obtain 2- = Y+ [] It follows from 1.Corollary 8.6 that . _ 2+ + 27 or, if 2- _ 91+ [1], then

Lemma 5.8: For J-bi-non-contractive operators membership of the class H is a (D-invariant property.

Let V be a J-bi-non-contractive operator, let V E H, and let 2'o be a neutral invariant subspace of V. By Corollary 5.4 dim 20 < oo, and therefore by virtue of 1.Corollary4.14 V9?o = moo. It follows from the result of Exercise 20 on 2.§4 that the operator f ,induced by the operator Vin the factor-space . = X'119o is J-bi-non-contractive. It is necessary to prove that VE H. By Corollary 5.3 there are invariant subspaces 9± E ,e fl h ± of the operator V which contain 2'o. But then 22± _ 9+I9o E,/u±(.°) fl h± (cf. Exercises 21, 22 on 1. §8), and 2'± are invariant relative to V. Let V± E ,if1 ± (.°) be arbitrary invariant subspaces of the operator V. Then .N"± = (x± E R± I R+ E 4'+) are maximal semi-definite invariant subspaces of the operator V, and therefore .N± E h±, which implies that ;V+ (=.4'±/9o) are members of the classes h± respectively. By Definition 5.1 then, VE H. Let 2' be a uniformly definite invariant subspace of a J-bi-non-contractive operator V, and let V9 = 2' if 2 C ,+. Without loss of generality we suppose that 2' C+ or 22 C W- depending on the sign of Y. Then W _ Y [ O ] Y[1] and relative to this decomposition the operators V and J can be expressed in matrix form: V=II Vii II?i=1,

V21=0,

0Ep(Vii),

and

J= II JiJII?!=1,

J21=0,

12=O,

and

if

2'C.?

or

ll=-Il2if22C.?-;

J22=

J*_

l

3 Invariant Semi-definite Subspaces

196 and

22 = J* = J22'.

If 2E .10-, then

follows from Exercise 34 on 2.§4 that

it

V22

is a

J-bi-non-contractive operator. Suppose

2'C.+, X2E2'

,

and

=-V-'V12x2.

x1

Then [ V22x2, V22x21 = [ V(xi + X2), V(xi + X2)] i [XI + X2, xi + X21

= [xl, xi] + [X2, x2] i [x2, x21,

i.e., V22 is a J22-non-contractive operator. Since 2[l] is invariant relative to

the J-non-contractive operator V` it follows that (V22)` (=(V`)22) is a J22-non-contractive operator, i.e., V22 is a J22-bi-non-contractive operator.

Thus, in both cases-whether 9 C ,JP + or 9 C 9--the operator V22 is J-bi-non-contractive.

Now suppose, moreover, that V E H. We verify that then V22 E H also. Suppose, for example, that 9 C :+, and that 2+ (E./tf+ ), existing by virtue of Corollary 5.3, is an invariant subspace of the operator V containing 9, and that 9+ = 9 [+] 9. .9 is completely invariant relative to V, so V2292 = 22 . It follows from 2+ E /ff + fl h+ that 9 E +ww) fl h+, i.e., the operator V22 has at least one invariant subspace from -&+ (91,W) fl h+. We now assume that V22,4-2 = T2+, and ./U= Lin[2'l.Az J. Then

A z E.alt+and

( . W ) and V,11'= IV. By Definition 5.1 A' E h+ and so -Y? E h+ also, i.e., all maximal non-negative invariant subspaces of the operator V22 belong to the class h+.

By Theorem 5.2 V22 E A+ and therefore V22 has at least one maximal non-positive invariant subspace (in 91±] and therefore also in .Y). Let 2- be such a subspace. Then 9+ = Y- 41 n Ytl1 (E.11Z+211) is an invariant subspace of the operator V`. Hence 2+ E h+, which implies (see Exercise 12 on 1.§8) that 9- E h-, i.e., V22 E H. A similar assertion in the case 2' C ?- is proved in the same way. Thus we have proved Lemma 5.9: Let 2' be a uniformly definite invariant subspace of a J-bi-noncontractive operator V, with V2 = 2' if 2' E let P, be the J-orthoprojector from . on to 9, and let V22 = (I - P') V(I - Pr) (I - Pr)'. Then V22 is an 40+,

(I - P,)*J(I - P,)-bi-non-contractive operator, and if V E H, then V22 E H also.

Definition 5.10: We shall say that a family of operators .1'= [ X] which commute with a J-bi-non-contractive operator Vo of the class H (i.e., 2

§5 Operators of the classes H and K(H)

197

XVo J VoX), with p(X) fl C+ it 0 for every X E ?' belongs to the class K(H) and we shall write E K(H). We remark that ;t' may consist of a single operator X and we shall then say that the operator X belongs to the class K(H) (X E K(K )). As well as the notation 2" E K(H) we shall also use the symbols -E K(H, Vo) to show precisely with which operator of the class H it is that .' commutes. Lemma 5.11: Let I,= (V ] be a commutative family of J-bi-non-contractive operators with 1' E K(H, Vo). Then when J * I (respectively, J ;d - I) the family 'o = 7/ U Vo has a non-trivial completely invariant non-negative subspace _T+ (respectively, invariant non-positive subspace Y_).

We consider first the case when the operator Vo has no neutral eigenvectors. Then, in accordance with Lemma 5.7, there is a single pair Y+ E .W and Y- E A 1 of subspaces invariant relative to Vo, and VoY+ = Y+ It

follows from the uniqueness of Y+ and 2- that Vf+ = Y+ and V2- C 2for any VE 11, i.e., 11/o2+ = 2+, 'Yo Y- C -T-. We put T+ = Y± We now assume that the operator Vo has at least one neutral eigenvector xo: Voxo = Xoxo. Then Ker(Vo - X01) is an invariant subspace of the family V. If

Ker(Vo - XoI) is degenerate, then, in accordance with Corollary 5.6, its isotropic part 4'00 is finite-dimensional. We consider the three possible cases:

a) IX01 < 1; b) IXoI> 1; c) IXo1. a) I X0 < 1. In accordance with Exercise 6 on 2.§6, Ker( Vo - XoI) is a

non-positive subspace, and therefore (see Exercise 17 on 2.§4) .4't is invariant relative to each of the operators VE Y Moreover, by "Q0 2.Proposition 4.14, V.,4"Oo = A "0,, since dim , l < oo. We put Y+ _ 9- = .410. b) I Xo I > 1. In accordance with Exercise 5 on 2.§6, Ker(Vo - XoI) is a non-negative subspace. It follows from Exercise 17 on 2.§4 that .%1.O is the

isotropic part of Ker [ Vo - (l/Xo)I] and therefore V`.iV°x. = 1 0o for all V E T. We again use Exercise 17 on 2.§4 and we obtain that V41 = ol '°o for all V E Y' We again put Y+ = Y_ = ,11'L c) 1 ao 1 = 1. with 19 2.§4, In accordance Exercise on Ker( Vo - XoI) = Ker( Vo - )%ol), and therefore Ker( Vo - X0I) is invariant relative to both V and V` for all VE 1'. Consequently, if yo E At, then [ Vyo, x] = [yo, Vo x] = 0 for all x E Ker( Vo - X01), i.e., V.4'00 is a subspace

in VO,. From 2.Proposition 4.14 and the fact that

.4"0o is

finite-

dimensional, we obtain that V.NO0 = 4'°, for all V E Y . Again we put We now assume that Ker(Vo - X01) is non-degenerate. In accordance with Corollary 5.6 Ker( Vo - XoI) is a Pontryagin space. From the consideration of the cases a)-c) we see that I Xo I = 1, and the family Y induces in Ker (Vo - XoI)

198

3 Invariant Semi-definite Subspaces

bi-non-contractive operators relative to the form [ , ] I Ker( Vo - )bI). By virtue of Exercise 2 on §4 we obtain that these operators have common non-trivial semi-definite invariant subspaces 2'+ and 2'_, and that (VI Ker(Vo-XoI))J'+=2'+. Theorem 5.12: Let a commutative family Y = ( V) of J-bi-non-contractive operators belong to the class K(H, Vo). Then the family Y o = Y U Vo has the property 4). Since the property (a family of J-bi-non-contractive operators is commutative) is 4?-invariant, it follows from Lemma 5.8 that the property (Y o belongs to the class K(H )) is also 4?-invariant. It therefore suffices to prove that every maximal completely invariant subspace 2'+ C _P++ U (B) and every maximal

invariant subspace 2'_ C .40 -- U (0) of the family i'o belong to tl+ and -lfrespectively. Let 2' be one of such subspaces. By Corollary 5.4 2' is uniformly definite, and therefore N' = 2' [--] 211" . let Pr- be the orthoprojector from . on to Y. We consider the family Y 02 = ( Vzz) U (V0)22 of V- Pr-)*J(I - Pr)-binon-contractive operators of the class K(H, (Vo)ZZ) (see Lemma 5.9), where

V22 = (I - P,)V(I - PA I

Yf11,

(Vo)22 = (I- P,')Vo(I - P1) I

x,[11

From the maximality of 2' and Lemma 5.1 applied to the family X02 we conclude that 9?[11 has the sign opposite to that of Y. In accordance with 1.Proposition 1.25 2 is a maximal semi-definite subspace. Corollary 5.13: A J-bi-non-contractive operator V of the class K(H) belongs to A+ fl A_ and has the property.$. This follows immediately from Theorems 5.12 and 5.2.

3

Let a J-bi-non-contractive operator V belong to the class K(H, Vo), Yo be the maximal invariant neutral subspace of the operators Vo and V, and let 2'+(E,//+) be invariant subspaces of these operators containing 2'o. Let 2+ = 2'o [ + y' _ 2' [ O ] 99_ be the decompositions of the subspaces 2+ into the sum of a neutral subspace Yo and uniformly definite subspaces 2±. Since 2+ E //+ fl h± we have ?o11 = Lin (2+, 2- ), and therefore, by virtue of the formula 1. (7.4) and 1.Corollary 8.16 "Y = 2'0 0+ (2'+ 4-

'-) O J?o.

(5.2)

Relative to this decomposition the operator V is represented by the matrix

V=

Vii

V12

V13

V14

0

V22

0

V24

0

0 0

V33

V34

0

V44

0

(5.3)

§5 Operators of the classes H and K(H)

199

We mention that in a number of cases later (for example, in the proof of Proposition 5.14) we shall make use of the condition VE K(H ) in order to have the expansion (5.3), and therefore it would be possible to demand the existence of the expansion (5.3) instead of the condition VE K(H ). However, in order to avoid awkwardness of presentation we shall not do this. 5.14 Let J-bi-non-contractive operator V belong to K(H), and let e E ap( V) fl T. Then the root lineal 99A V) is a subspace, and it splits up into a

J-orthogonal sum 2e( V) = Nf [ + ] .Jfe, where dim ./V, < oo, VA', = .ib"ei . Ife C Ker(V - cI) and -ffe is a projectionally complete subspace or, in particular, lfe = (0). We consider the decomposition (5.3) of the operator V. The subspace lfe = tee( V) fl Ker V12 + f(V) fl Ker V13 is regular and moreover i = 1, 2, 3. The operator V22 is non-contractive Vie( V) fl Ker V,; C Y4

relative to the scalar product [x, y] (x, y E 2+ ), and V33 is non-expansive relative to the scalar product - [x, y] (x, y E ?_ ). Therefore (cf. Exercise 23 Ker( V;, - cIi), i = 2, 3, which is equivalent in the present on 2.§4) _T,( case to the inclusion .ite C Ker(V- eI). Since by virtue of Exercise 19 on 2.§4 /f, C Ker(Vc - eI), and qp(V) = V, [+] .ife, where .Ne =X21] fl Y,( V), it follows that .-/V, is invariant relative to the operator V. By construction dire [1] Ye(V) fl Ker Vl; (i = 2, 3).

It remains to verify that IV, is finite-dimensional, from which it will also follow that it is completely invariant. We assume the contrary: '4', is an infinite-dimensional subspace, and ?o, 2'+ and Y_ are the same as in the decomposition (5.2). Since dim(.?o O+ J-Wo) < ao, so a', fl ('+ + Y_) is an infinite-dimensional subspace, and therefore at least one of the subspaces fl Y_ will be infinite-dimensional. Suppose, for example, .'1 fl _r+ or that dim Fe fl Y+ = oo Since dim 2+Ker V,2 < oo, it follows that .4l"f fl Ker V12 ;e (0), and we obtain a contradiction with the fact that .4l", [1] 2e(V) fl Ker V12.

From Corollary 5.6 and Proposition 5.14 we obtain Corollary 5.15: It under the conditions of Proposition 5.14 the operator V belongs to H, then, when dim Y4 V) = co, .,lff, is a Pontryagin space with a finite number of either negative or positive squares.

4 We now consider a family 1= (A) of maximal J-dissipative operators of the class K(H) and we investigate for it the questions of the existence of maximal semi-definite invariant subspaces. Theorem 5.16: Let .4 = (A) be a set of maximal J-dissipative operators, let

.-c/EK(H, Vo), and let .'+ (C.?+) be invariant subspaces of the family ,V U Vo with Y+ completely invariant relative to Vo; and suppose there is a

3 Invariant Semi-definite Subspaces

200

point XA E p(A) fl C+ such that XA, XA E p(A 1 9?+) and XA E p(A I 9_ ). Then,

if the resolvents of the operators A E ' commute pairwise, there are extensions ?+ E .,tl± of the subspaces 2'± which have the same properties.

D We consider the family of operators r = (V = Kx(A) I A E d). In accordance with 2.Theorem 6.13 the V are J-bi-non-contractive operators. It follows from the inclusion 4 E K(H, Vo) that r' E K(H, Vo). Since XA, XA E p (A I '+ ), the subspace 2'+ is completely invariant relative to 'V, and the condition XA E p(A I T_) implies the invariance of 2'_ relative to 1. Consequently, by virtue of _Theorem 5.12, there are k+ E ill± fl h- containing U Vo. We now Y+ and such that VV+ = 2'+, V2'- C 2'_ for all V E I o = prove that k+ are invariant relative to .d, and that XA, XA E p(A I 2'+) and

XA E p (A I 2'_ ). To do this we note that 2'o = .'+ n 2- is an invariant finite-dimensional neutral subspace of the family 11, and therefore, by virtue of Exercise 13 on 2.§6, Yo c InVA I A Ed) and .d2'o C 2'o. We bring into consideration the i-space = 2'011/2'o and the families of operators acting in it gel = (A) and Y'o = (V) U Vo generated by the families and 1/0. Moreover, lA = (qA n 201")/2'o, XA E p(A) and KXA(A) = KxA(A). We note that because k± are maximal semi-definite the equality (.+ + 2'_ ) [1] = 2'o follows, and since 2± E h±, so (see Exercise 19 on 1.§8) 2011 = 2'+ + ,'_, and therefore

/2'o + 2'_ /1o. Relative to this decomposition of f the operators FE 'To have the diagonal form

Y=

V,1

0

0

V22

and therefore

- ( AK-' A

which implies that -ql+

)-

V

II

KXA'(VI I) 0

0

KXA'(V22)II'

are invariant relative told and that XA and XA belong

to p(A I 2'+/2o) and XA belongs to p(A 2'_/2o). It only remains to use the fact that XA, XA E p(A 12'o).

This theorem enables us to prove the following assertion which will be useful later.

Theorem 5.17:

Let A E K(H, Vo) be a maximal J-dissipative operator, let

be a J-space, where Kero(A - XI) is the isotropic part of the subspace Ker(A - XI), and let Vo be

X = X E pp(A ),

let .W' = Ker(A - X)/Kero(A - XI)

the J-bi-non-contractive operator induced in .' by the operator Vo. Then VoEH.

In. accordance with 2.Corollary 2.17 Ker(A - XI) = Ker(A ` - XI), and therefore Ker(A - XI) is a common invariant subspace of the operators Vo

and V. Hence, it follows, in particular, that Kero(A - ceI) also is their common completely invariant subspace. We conclude from 2.Theorem 1.17 that Vo is a f bi-non-contractive operator. We now verify that Vo E H. Since

§5 Operators of the classes H and K(H)

201

Vo E H it follows that it is sufficient to find at least one maximal non-negative

invariant subspace of the operator Vo. Let Y+ be the common maximal non-negative invariant subspace of the operators A and Vo (we know that there is such a subspace from Theorem 5.16). We consider the subspace P), = Y+ fl Ker(A - XI). We first assume, and later prove, that the deficiency (see 2.Theorem 4.15) def -Cvx of the subspace

'x = Lin(2\ Kero(A - XI)J/Kero(A - XI) is finite. By virtue of Exercise 4 on §4 the operator Vo has at least one maximal non-negative invariant subspace, and therefore, as proved above, Vo E H. It remains to verify that def _4 < oo. We consider the decomposition of the root subspace 2' (A) = 4'x [+] / , where /t"x and /lh, are invariant relative to A, dim ,/t'x < co, and M), C Ker(A - XI). The existence of such a decomposi-

tion follows from Proposition 5.14 and 2.Theorem 6.13. We assume that def 2a = co. Since dim ,/VX < co, there is an xo E Ker(A - XI) which

J-orthogonal to 2a(A) fl Y+. In accordance with 2.Corollary 2.18 xo is also J-orthogonal to all the subspaces 2',(A) fl Y. Let Yo be the isotropic part of the subspace 2+. Since dim Yo < oo, we have

is

We consider the 2 0 C Lin[X,(A) fl Y+ j it E a(A)], and therefore J,-subspace , = Lin[Y+, xo)lYo . Since 2+ E h + and xo 0 Y+, it follows that .°i is Pontryagin space with one negative square. Since 20 and Lin[,?+, xoI 9,1.'

are invariant relative to A it follows that A induces in, a J1-dissipative operator A. Since 20 is the linear envelope of the isotropic parts of the subspaces ? (A) fl 2'+(µE ap(A)), the root subspaces of the operator A are

positive-which contradicts Corollary 2.9 and Theorem 1.13, according to which the operator A must have at least one non-positive non-zero eigenvector.

Corollary 5.18:

Under the conditions of Theorem 5.17, if Ker(A - XI) is not semi-definite, then the operator A I Ker(A - XI) has semi-definite invariant subspaces Ya of both signs, which are maximal in Ker(A - XI). Moreover,

for all X = except possibly a finite number of points we have !a + 4- = Ker(A - XI), and for the remaining X E P we have dimKer(A - XI)/2' ); + 91Z < co.

Let Ker(A - XI) be not semi-definite, let !a be the semi-definite invariant subspaces, maximal in e, of an operator Vo belonging to the class H, and let 2a be their generating subspaces in Ker(A - XI). The maximality of .-a in .)Y implies the maximality of YK in Ker(A - XI), and the invariance of ._a

relative to Vo implies the invariance of 9a relative to Vo. Sincea E h±, it follows (see Exercise 19 on I.§8) that dim dim M fl a < oo, which in turn implies that (dim .i(/2 + jla =) dim Ker(A - XI)/ft + 91a < oo.

From the J-orthogonality of Ker(A - X1) and Ker(A - µl) when X µ, X, µ E ER, and from their invariance relative to Vo, Vo (EH ), it follows that only

3 Invariant Semi-definite Subspaces

202

a finite number of the subspaces Ker(A - XI) and sa can be degenerate. Moreover, again from the fact that Y,; E h- we obtain that dim Ker(A - XI)/f)+, + 91a = dim Ya n z- - dim Ko(A - XI), and the equalities

dim Ker(A - XI)/2 'a + £a = 0 and -Ta + Ya = Ker(A - XI) are equivalent.

5

Let j11= [ U) be a commutative group of J-unitary operators, and

1 E K(H, Vo). If Vo is also a J-unitary operator, then, by carrying out an argument similar to that used in the proof of Theorem 5.12, we can satisfy ourselves that the family -lo = %' U Vo has the property i) ['l. But if Vo is not a J-unitary operator, we can at present assert only that the family -11o has the property . However, in any case the group W does have the property JlJ . For, let Y+=moo[+]Y+E..rr+n h+ (WO C.-Po,c+C ++U[O)) be an invariant subspace of the group X71. Then Y_

=,+ [iJ

= So [+].S_ E _rr- n ..*,-

is also an invariant subspace of this group. Without loss of generality we can suppose that 2'+ C W± and £'o J. (2+ [ @] S-) Therefore in the decomposition (5.2) of the sRace ye all the components are pairwise orthogonal to one another and (moo + JIo) [1] (Ib+ [ (@] Y_ ). Let ./V+ (C P ±) be arbitrary maximal invariant subspaces of the group %1, with A + [1] /I"_, i.e., A'-) is a dual pair. We verify that .4'+ E To do this we consider the subspaces .4'± = ! "± n 2o1J. This too is a dual pair, invariant relative to X11, 1"± and dim I /I"± < dim A < co. Together with will (,/I"+' , also be an invariant dual pair if /I"+' = Lin [ 'o, .4 + ). We consider the f-space _ 0 IYo and the group '11 induced in it by W. Since Ib± C ,Y±, we have = I+ [ O] 2-, 11 + = J'±, and therefore %1l is a group of operators which are simultaneously J-unitary and unitary. By Theorem 4.1 '11 has the property J1J, and therefore (see Exercise 7 on 1.§5) there is an ..!"+ E,11+ such that is a dual pair invariant relative to 1, and moreover /I"+' C,, j7+. The subspace A", = Lin [.A"+, .'I "+) is invariant relative to -V1, is J-orthogonal to A" and contains at least one maximal non-negative subspace (/l "+ ). By construction it is a W(')-space with x (
quently (cf. 1.Theorem 10.2 and I.Proposition 8.18) every non-negative subspace which is maximal in A', is also maximal non-negative in -Y. _ We verify that there is in ./l ", a maximal non-negative subspace containing .4"+ and invariant relative to W. To do this it suffices to note that in ./I.+

the G(' -space .4{ ', = ,l -,/.'! "o (where .A! "o is the isotropic part of ./1', ) the group

X11,, induced in 4", by the group -11 1 A",, has (by Corollary 4.13) a maximal

§5 Operators of the classes H and K(H)

203

non-negative invariant subspace containing Lin(N+, /V_ ), and then to make use of the second part of the proof of 1.Theorem 10.2. and dim .A"_/.iV_ < co it follows that Since .il"+ C .A [l "[1] < co. Since (IF, n 4"I1.A _) is an invariant dual pair of

dim,il'+n

the group

! and contains

def /V+ < oo. By Theorem 4.14 Thus we have proved

A'_), we have A"+ n 4 11 = .4 + and A"±

E U-.

Theorem 5.19: A commutative group of J-unitary operators belonging to the class K (H) has the property4 . In this paragraph we describe the structure of the spectra of J-unitary and J-self-adjoint operators of the class K(H ).

6

Let U be a J-unitary operator and U E K (H, Vo ), and let x," = max(dim `moo I Yo C .00, Vo2o = Yo}. Then the non-unitary spectrum of

Theorem 5.20:

the operator U consists of a finite number x, (>,0) of pairs of normal eigenvalues disposed as mirror-images relative to the unit circle T, and ap(U) n T contains a finite number x2 (,>0) of points to which correspond the non-prime elementary divisors, and x, + X2 < xv

By virtue of Corollary 5.13 U E A + n A_ , and from Remark 2.4 and 2.Corollary 5.13 we conclude that the non-unitary spectrum of the operator U consists of normal eigenvalues. We use Exercise 6 on 2.§6 and obtain that 21, = Lin(Ya(U) I I X I > } is a neutral lineal. Since Vo2', C 91,, we have dim 2'i = xi < xv0. Since the spectrum of a J-unitary operator is symmetric relative to the unit circle, the first part of the theorem has been proved. Let -9 = (e E T I UI Y, (U) has non-prime elementary divisors). Then, by virtue of Exercise 14 on 2. §6, Ker(U - el) is a degenerate subspace; let Y0 denote its isotropic part. Since Ker(U - el) is invariant relative to Vo and Vo, we have VoYof! = Yo, and therefore '2 ° Lin (Y° I C E e) is a neutral lineal invariant relative to Vo. Consequently dim Y2 = x If < x v,,, and so f consists of a finite number x2 of points and x2 < x2'. Since Lin(2,, Y2) is also a neutral subspace invariant relative to Vo, we have x, + x2 < x i + xz < X V,,Corollary 5.21: x, ,

Let A be a J-self-adjoint operator and A E K(H, Vo), and let

be the same as in Theorem 5.20. Then the non-real spectrum of the

operator A consists of a finite number x, (,> 0) pairs of normal eigen values symmetrically situated relative to IR, and ap(A) n IR contains a finite number x2 (>, 0) of points to which correspond to non-prime elementary divisors, and x, + x2 < xV,,-

It follows from the definition of the class K(H) that the Cayley-Neyman

transform K,,(A) of an operator A belongs, when X E p(A), to the class

3 Invariant Semi-definite Subspaces

204

K(H, Vo) whenever A does. It remains only to apply Theorem 5.20 and II.Corollary 6.18. Thus, if a,, X1; X2, X2; . . . Xx,, Xx, are the different non-real eigenvalues of a J-self-adjoint operator A E K(H ), then "I

.f = [+] (Ya; (A) + _1rs,(A )) [+] 4e',

(5.4)

i=i

where Y' is the J-orthogonal complement of E;,k , [+] (14 (A) + 1a;(A)), and the operator A is expressed relative to (5.4) by a diagonal matrix which we write symbolically as x,

A = [+](A),;+Aa) [+]A',

(5.5)

i=1

where A,, = A I Y, (A), and A' = A I .W', and moreover the spectrum of the operator A' is real, and a(A,,) _ [µ) (µ = X1, X,; X2, X2; ...; Similar expansions ,W=

[+](I%,(U)+tee,i(U))[+]

(5.6)

i=1 x,

U = [+] (Ue; + Ue, ') [+] U'

(5.7)

also hold for J-unitary operators U of the class K(H ), where [ei, e;

the different non-unitary eigenvalues of the operator

are all

U,

E [ ei, ti-'), and the spectrum a(U') is unitary.

In particular, the expansion (5.6) and (5.7) are valid also for J-unitary operators of the class H. 7 In conclusion K (H ).

we give some examples of operators of the classes H and

Example 5.22: Every J-non-contractive or J-bi-non-expansive operator in Ux belongs to the class H (cf. Definition 5.1 with Corollary 2.9).

V be a J-bi-non-contractive operator and let V = V' + V", where V" = I I V; I I;;=, is a diagonal operator relative to the .,Y-; let p (vii) fl P0/12) = 0, and V" E Y.. Then decomposition Example 5.23: Let

VE H.

We conclude from the representation V = V' + V" that V E A+ fl A_ . E7 Consequently, by virtue of Theorem 2.8, it has at least one pair &+ and !- of maximal semi-definite invariant subspaces. Let K± be the angular operators of these subspaces. In accordance with Lemma 2.2

K+ V - V22K+ = - G v°(K+) and

V;1K- - K- V22 = G V-- (K- )

§5 Operators of the classes H and K (H)

205

Since p(vii) n P (V22) = 0, we obtain from [VI] that these equations can be rewritten in the equivalent form

K+= -

(V2'2 - XI-)-'G v.'(K+)(Vii - µI+)+'

1

47r

K-

(Vl' -µI+) 'G = - 47r12

r,

µ-

r,..

dX dµ

(5.8)

)(Vz2 - XI-)-' dX dµ

(5.9)

where IF v;, and r v j, are non-intersecting Jordon contours surrounding a( VI l ) and a(V'2) respectively. Since V" E .99., so G:.. (K±) E 9 9., and therefore we

obtain from the equations (5.8) and (5.9) that K± E 91m. It only remains to

notice (see Exercise 18 on I.§8) that this is sufficient for the inclusions 97+ E h±.

Example 5.24: An amenable uniformly bounded group W of J-unitary operators belongs to the class K (H ). As the operator Vo with which all the operators U E Qi will commute we may take (see Exercise 1 below) Vo = p+ - p-, where P+ are the Jorthoprojectors on to the components of the maximal uniformly definite dual

pair (97+, 97-) invariant relative to °?l (the existence of such a pair is guaranteed by II.Theorem 5.18).

Exercises and problems 1

Verify that the operator JE H.

2

Prove that every uniformly J-bi-expansive operator belongs to the class H. Hint: Use Theorem 2.1.

3

Prove that a J-unitary operator U belongs to the class H if and only if there is a canonical decomposition W' = ,Yl [+] .,Y1- such that the angular operators (with respect to this decomposition) of all invariant subspaces 99 ± E //± of the operator U are completely continuous (Azizov [9]).

4

Prove that, under the conditions of Proposition 5.14, if V E K (H) and if di (f), d2(f), ... , dr,(e) are the orders of the elementary divisors of the operator VI -We, then [di (f ) eET i=t

where x,.,,

2

is the same as in Corollary 5.5. In particular, if V is a J-unitary

operator, then r,

di(f)

eET i=i

(Azizov: cf. Usuyatsova [2] ).

2

x>1

3 Invariant Semi-definite Subspaces

206

Hint: Similarly to that was done in proving 2.Theorem 2.26 show that there is in M' (V) a E;=, ([ d;(c)]/2)-dimensional neutral subspace invariant relative to V0. In the case of a J-unitary operator use Theorem 5.20. 5

Formulate and prove an analogue of Proposition 5.14 and Exercise 4 for maximal J-dissipative operators and, in particular, for J-self-adjoint operators of the class K(H) (for the latter taking Corollary 5.21 into account).

6

Prove that is a J-bi-non-contractive operator Vo belongs to H, and if Yo is its finite-dimensional

non-degenerate

invariant

subspace,

then

the

operator

Vo I 91J1J E H (Azizov).

Hint: Use Lemmas 5.8 and 5.9. 7

Prove that in the expansion (5.7) U' E K(H) (Azizov). Hint: Verify that if UE K(H, Vo), then Vo.Yt" C .W', and use the result of Exercise 6.

8

Prove that if Uo E H is a J-unitary operator and if all its root subspaces 2'((U) with eI=1

are non-degenerate, then the space ."' in the expansion (5.6) can be

represented as the J-orthogonal sum of subspaces invariant relative to Uo and any operator community with it: k

.W' _ [+] 4;(Uo)[+]

(5.10)

"

where the ('r are the eigenvalues of the operator Uo to which corresponds at least

one non-trivial neutral vector (i = 1, 2, ... , k), and W" is the J-orthogonal complement to the first k terms of the expansion (5.10) (Azizov). Hint: Use Theorem 5.20 and Corollary 5.15. 9

Let eW = ( A) E K (H) be a family of maximal J-dissipative operators, let a/ E (L)

and the resolvents of the operators of this family commute pairwise, and let 1/'+ c p+ n (n VA I A E . /) be an invariant subspace of the family ._d. Prove that then there is a subspace 5C'+ D Y+ : 9+ E ..It+, 9+ C (n 1'A I A E,V), d9+ C

'+

(Azizov [9] ).

Hint: Use Theorems 1.13 and 5.12. 10

Let :-V = (B) be a commutative family of J-bi-non-contractive operators containing an operator Bo E H, and let W = (A) be a family of J-bi-non-contractive operators which has the property that AB = BA for any A E ', BE -V and for arbitrary A 1, A2 E . / there is a B12E . such that Al A2 = B12AZA I. Prove that then the family ..el U .-3 has the property 44 (Azizov and Dragileva [1]).

11

Prove that a metabelian group of J-unitary operators of the class K(H) has the property 4i [11 (Azizov; see [IV]).

12

Let Y = (V) be a scalarly commutative family (i.e., for every V1, V2 E Y there is a constant X12 such that V1 V2 = >112 V2 V1) of J-bi-non-contractive operators containing

an operator Vo E H. Then Y has the property 4 (Azizov). Hint: Use the scheme of the proof of Theorem 5.12. 13

Prove that if, under the conditions of Exercise 12,

Y

is a group of J-unitary

operators, then it has the property 4 L ] (Azizov [9], Khatskevich [4], cf. Helton [2] ). 14

Prove that a bounded J-selfadjoint operator A belongs to the class K(H) if and only if it has at least one invariant subspace 9+ E ,/!+ n h+ (Azizov [13] ).

Remarks and bibliographical indications on Chapter 3 15

207

Let A E K(H) be a bounded J-selfadjoint operator, and let 2' be a finite-dimensional non-degenerate subspace invariant relative to A. Prove that A I 2' E K (H) (Azizov).

Hint: Let M'+ E .:#+ fl h+, AY+ C 2+. Verify that det(2+ fl 2'W) < oo, use Theorem 4.14 and Exercise 14. 16

Prove that if V is a J-bi-non-contractive operator of the class H, if Se is

its

projectionally complete invariant subspace, and if the operator V 191 has a maximal (in 2') completely invariant non-negative subspace 2+, then V 199 E H (Azizov). Hint: Use Corollary 5.4 and Theorems 5.2 and 2.8. 17

Let the conditions of Problem 16 hold, and let P be the J-orthoprojector on to Y111 . Prove that the operator PV I M'1' E H and that it is P*JP-bi-non-contractive (Azizov).

Hint: Consider the subspace 2'

E tl+ : 2'+ C 2+ V2+ = 2+ and prove that P1'+ E h+, (PV 12111 )P2+ = P2+ and that V` I 2'tll is a P * JP-bi-non-contractive operator.

Remarks and bibliographical indication on chapter 3 §1.1-1.3. In the formulation of Problems 1.2, 1.3, 1.6, 1.7, 1.10, and 1.11 their sources were indicated. To what was said there it should be added that S. L. Sobolev proved that a 7r-self-adjoint operator in II1 has a non-negative eigenvector. The examples 1.4 and 1.8 were constructed by Azizov. J-selfadjoint operators A with J' C iA were first considered by Langer [2], [3] (see Definition 1.5). The Remark 1.12 is also due to him (see Langer [13]. The reduction of Problem 1.10 to Problem 1.11 was borrowed from Phillips [3]. §1.4. Theorem 1.13 was proved by Azizov [9] §1.5-1.6. The definitions and propositions are due to Azizov. We mention that the idea of formulating Lemma 1.20 in terms of `a complex J (' (see also 4.Lemma2.5 below) was borroed from [XI]. §2.1. Theorem 2.1 is due to Azizov (see Azizov and Khoroshavin [1]). The

existence of the property c for a uniformly J-bi-expansive operator was proved earlier by a different method in [VI]. Similar results for doubly strict focusing plus-operators (cf. 2.§4) in the cases of real and complex J-spaces by the methods of compressed mappings in the metric mentioned in the Exericses on 1.§8 were obtained by Khatskevitch [6], [11], [15], A. V. Sobolev and Khatskevich [1], [2], and in the case of real II earlier by Krasnosel'skiy and A. V. Sobolev [1]. §2.2. Lemma 2.2 was established by M. Krein [4]. Regarding Definition 2.3, see Azizov and Khoroshavin [1] and also Azizov and Kondras [1]. In the last paper the Remark 2.4 was also made, a remark stimulated by Noel's article [1].

§2.3. Proposition 2.6 was borrowed from Langer (see [XVI] ). Theorem 2.8 is due to Azizov and Khoroshavich [1]. Its first part was essentially proved earlier by I. Iokhvidov [10]. The remaining assertions are encountered in particular cases and in parts earlier in the papers of M. Krein [4], [5], [XIV], Brodskiy [1], Langer [l]-[3], Azizov [4]-[8], Azizov and E. Iokhvidov [1],

208

3 Invariant Semi-definite Subspaces

Azizov and Usvyatsova [1], I. Iokhvidov, M. Krein, and Langer [XVI]. The

same also applies to Corollary 2.9. We remark that our proof of these propositions is based on the application of a device of M. Krein [4] of approximating to a J-non-contractive operator V by uniformly J-expansive operators VI,,. Corollary 2.10 was formulated by Azizov. §3.1. Krein-Shmul'yan linear-fractional transformations were first brought into consideration by M. Krein [4]. They and their generalisation were systematically studied in the articles of M. Krein and Shmul'yan [4], Shmul'yan [2], [6], [8], Larionov [4], Helton [1]-[3], Khatskevich [5], [13], [14], and others. Propositions 3.1 and 3.4 are due to M. Krein [4], [5]. §3.2. Theorem 3.6 plays in our exposition the same role as was played earlier in these questions by Tikhonov's principle and its generalization given by I. Iokhvidov [10]. Theorem 3.8 is a natural generalization of the corresponding propositions of M. Krein [5] and I. Iokhvidov [10]. Its simple proof given here, based on Glicksberg's thgeorem, is by Azizov. §3.3. Theorem 3.9-a direct consequence of Theorem 3.8-was formulated the first part was proved by Wittstock [1] and, essentially, earlier by I. Iokhvidov [10]; the second and third parts were obtained by M. Krein [5] (cf. Langer [3] ). Corollary 3.10-see M. Krein [5], I. Iokhvidov [10], Whittstock [1], and Brodskiy [1]. Corollary 3.11 was noted by Azizov. Theorem 3.12 is due to Langer [3], M. Krein [5]; the short proof of it in the text is due to Azizov. §3.4. Lemma 3.13 and Theorem 3.14 were proved by Azizov. Earlier results close to Theorem 3.14 were published by Larionov [4] and Khatskevich [5]. §3.5. Theorem 3.20 and Corollary 3.21 follow from Wittstock's results [1]; Azizov gave the proofs given in the text. §4.1. Theorem 4.1 is due to Phillips [3]; the proof of it given in the text is due to Azizov. §4.2. The results in this section are due to Azizov. Lemma 4.4 seems to be well-known. Theorem 4.5 in the case where W is commutative constitutes the main results of Phillip's work [3]. As Shul'man noted, Corollary 4.6 is also true without the requirement for %' to be commutative. §4.3. The idea of applying Theorem 4.7 in the questions examined here is due to Naymark [2]. Esentially Corollary 4.8 is also due to him. Corollary 4.9 was obtained by Larionov [2], [3]. Example 4.10 was constructed by Azizov. §4.4, §4.5. Theorem 4.11 was proved by Azizov [6]. Corollary 4.12 was proved earlier by Naymark [2]. Corollary 4.13 is due to Azizov. Theorem 4.14 is also due to him. and proved here by Azizov. In the case when V12 E .9

§5.1-5.4. Helton in his article [2] considered a commutative group of J-unitary operators containing an operator Uo which has at least one maximal

non-negative invariant subspace and the angular operator of each such subspace is completely continuous. This provided the stimulus for the introduction by Azizov [9] of operators of the class H and K(H ). All the results of these sections, except Proposition 5.14, are due to Azizov. Propo-

Remarks and bibliographical indications on Chapter 3

209

sition 5.14 was mentioned in Usvyatsova's paper [2]; the proof in the text was given by Azizov. §5.5. Theorem 5.19 (see also Exercise 13 on this section), which is due to Azizov [9] and Khatskevich [4], generalizes a corresponding result of Helton [2]; in this paper the property 4 AI is proved by a different method. §5.6-5.7. The results in these sections are due to Azizov.

4 SPECTRAL TOPICS AND SOME

APPLICATIONS

In § 1 the concept of a J-spectral function is introduced and some classes of operators for which it exists are distinguished. Some of its applications to the solution of problems of invariant subspaces for families of `definitizable'

operators and to the construction of the (J1, J2)-polar decomposition of (Jl, J2)-bi-non-expansive operators are also indicated. §2 deals with questions of completeness and basicity of systems of eigenvectors and principal vectors of J-dissipative and, in particular, J-selfadjoint operators. In §3 we give some examples and applications.

§1

The spectral function Let E be a homomorphism mapping a ring e?, generated by certain

1

intervals of the real axis and containing the interval (- oo, oo ), into a set of

J-selfadjoint projectors E(A) acting in a J-space ., and let E(0) = 0; E(-oo,oo)=I; E(Af10')=E(0)E/0'); E(DUA')=E(0)+E(0') if 0 fl 0' = 0 (0, 0' E W). We denote the carrier of this homomorphism by a(E). A point X E a(E) is called a point of definite (positive or negative) type if

there is an interval 0 E 41 containing X and such that E(A).Y( C ,?+ or E(A).Y( C .40 - respectively. The sets of points of positive and negative type are

denoted by a+ (E) and a- (E) respectively. Let s = {ai )1 be a finite set of real points, and let 9t = .91(s) be the ring generated by the intervals for which points from s are not boundary points. Definition 1.1:

The homomorphism E described above, defined on 5?(s) is

called a J-spectral function with a set s = s(E) of critical points if, when 210

§1 The spectral function

211

Xo, µo 0 s(E), µo < Xo, and µ 1 µoi the strong limit of the operators E((µ, X0] ). exists and it coincides with E((µo, Xo] ). Definition

1.2: A point a E s(E) is called a regular critical point of

a J-spectral function E if the strong limits s-limx ,,E((- oo, X]) and s-limaiaE((X, oo)) exist. Otherwise a is called a singular critical point. If s(E)

contains no singular critical points, then E is called a regular J-spectral function.

Definition 1.3: A J-spectral function E with a set s(E) of critical points is called an eigen spectral function of a J-selfadjoint operator A if:

1) AE(A) = E(0)A; 2) a(A I E(A)M) C 0; 3) when 0 E 91(s) and o fl s(E) = 0 we have AE(0) = Jnt dE,, where the integral converges in norm if A is finite, and strongly if 0 is infinite. The set s(E) is called the set of critical points of the operator A and is denoted

by s(A), i.e., s(E) = s(A). 2 We now indicate some classes of J-selfadjoint operators which have a J-spectral function. One of them is the class of operators K(H) and, in particular, H (see Exercise I below). `Definitizable operators' form another class.

Definition 1.4: A J-selfadjoint operator A with p(A) ;d 0 is said to be definitizable if there is a polynomial p(X) such that [p(A)x, x] > 0 when X E cAk, where k is the degree of the polynomial. In particular we observe that a J-non-negative operator A with p (A) 0 is definitized by the polynomial p(X) = X. We shall not stop now to prove that definitizable operators have a J-spectral function; this result is formulated later as an Exercise. Instead we shall prove the key theorem in the theory of definitizable operators.

Theorem 1.5 Let A be a bounded J-non-negative operator. Then it is possible to set up in correspondence with every real number X ;d 0 one and only one J-selfadjoint projector EE, such that the function X E,, satisfies the following conditions: 1) if X 5 it, then ExE,, = EEx = Ea;

2) if X <,u < 0, then [Eax, x] > [Ex, x], and if µ > X > 0, then [E,,x, x] <, [Ex, x] for all x E *9;

3) ifx< -11A1l, then E,,=0, and ifX Ail, then E),=T; 4) if X ;d 0, then the strong limit s-lim,,lxE = E),+o exists and coincides with Ex;

5) If T is a bounded operator which commutes with A, then TEa = Ex T-,

4 Spectral Topics and Some Applications

212

a(A I E>,.) spectrum 6) the a(A I(I- EE). ) C [X, co).

lies

(-oo, X]

in

and

v dE2, converges in the strong operator topology as an Moreover, JIIAII -IIAII o with a singular point X = 0, and the operator improper integral S=_ A - JIIIIAIII - o

v dEE is a bounded J-non-negative operator, and S2 = 0,

SE), = E,,S = 0 when X < 0, and S(T - E),) = (T - E>,)S = 0 when X > 0.

We put B = JA. This is a non-negative operator, and therefore the operator C = B 1/2 JB 1/2 = C* satisfying the condition CB 1/2 = B 1/2A is 1/2 112

properly defined. Since II C II 5 II B sion of the selfadjoint operator

= II B II = II A II, the spectral expanbe written in the form

C can

is the spectral function, continuous C = 11-I IIIIAII - ox d F,,, where (F,) on the right, of the operator C. We put C,, = C I Fi,M when X < 0, and Ca = C I (I - Fx).f when X > 0, and E>, = JB1/2C 1 FF,B'/2 when X < 0 and I - Ea= JB1/2Ca'(I- F),)B1/2 when X> 0. 1) The operators Ex, are bounded for every X # 0. Moreover, if X < 0 Fx (respectively, Ca' (I - F)')) are selfadjoint

(respectively, X > 0), then

operators, and therefore the Ex are J-selfadjoint

with X ;4 0. Since

CCx ' Fx = Fa when X < 0, and CCa ' (I - Fx) = I - F>, when X > 0, it follows that ExE,, = EE), = Ex when X <,u. Hence, in particular, we conclude that the Ex are projectors.

2) If X
1

FB 1/2x, B 1/2x) - (Ca 1 F,B 1/2x, B 1/zx) f µ 1 dFa(B'/2x, B112x) -Jav

0

for every xE e.

Similarly it can be verified that [EE,x, x] < [Ex, x] when 0 < X < µ. Assertion 3) follows directly from the definition of the function X - Ea. 4) Let X <,u < 0. Then II E,,x- Eax ll2 = II JB112(Cµ' F,, - Ca 1 Fi,)B112x II2

II B II :

d(F,,B1/zx, B1/zx) 5 II B II v2

-

II(Fµ - Fa)B1/2x 11 2,

and since by hypothesis (Fxj--- is a function continuous on the right, s = lim1,1E,, = E. The assertion is verified similarly when 0 < X < µ.

5) Let T be a continuous operator and let TA = AT. We verify that TEa = E1,T when X < 0. By definition, E1, = JB'/2 j''- 1A11 -0 1/v dFFB1/2 when

X < 0. Let 2 (Pn)' be a sequence of polynomials with

sup (Pn(X)IXE [-IIAII,IIAII)<M
and

lim pn(v)=1 if -IIAII
n-w

n-ao

§1 The spectral function

213

Then

s- lira JB'/2pn(C)BV'12 = E. n-m Since

JB112C"'B112 = A"'+' (m=0,1,2 ....), we have TJB''2pn(C)B1/2

= JB1/2pn(C)B1/2T, and therefore TES, = E1, T. Similarly one verifies that TEa = E),T when X > 0. In particular, AEI, = ExA.

6) We observe that, for any bounded operators T1 and T2 the sets a(TiT2) U (0) and a(T2T1) U (0) coincide. This follows immediately from the formula (T2T1 - XI)

(T2(T1T2 -

XI)-'T1 - I)

which holds when 0 * X E p (Ti T2 ). Consequently, since

AE1,= JB'i2F',Bv2 and

A(I-E),)=

JB1,12(I-

F)JB1,12

(X ;4 0)

we have

a(AEx) U (0) = a(B1/2JB1/2F),) U (0) = a(CFx) U (0), and

a(A(I - Ex)) U (0) = a(B1/2JB''2(I - Fi,)) U (0) = a(C(I - F,)) U (0). Hence

a(A I Ea.') C (0) U (- oo, X], a(A I (I- E),), W) C (0) U [X, co).

It remains to verify that if X < 0, then 0 0 a(A I E), ,Y), and if X > 0, then 00 a(A I(I - Ea).W). We shall verify the first of these statements; the second is

verified similarly. Suppose that X < 0 and 0 E a(A Ex'). Since A I Eat is a non-negative operator relative to the form [ , ] I E), and E),.*' is a Krein space, it follows by virtue of 2.Corollary 3.26 that a,(A I E),,W) = 0, and therefore 0 E o,, (A I E),.-W) U a, (A I E>,.W). In both cases there is a sequence (xn) E such that 11 Eaxn 11 = 1 But then and AE>,xn - 0. BV2AEExn = (Bv2E',X = C,FXB'12Xn - 0 also. Consequently Exxn 0 also-we have a contradiction.

We now verify that the function X -+ E), satisfying the conditions 1)-6) is uniquely defined. Let X - Ex' be another such function with X < µ < 0 or 0< X < µ. It is clear that E a = E;E,+ E),(I - En). By condition 5) the operators E, and E,, commute. Consequently, Ex(I - E,) is a J-orthogonal projector on to the subspace Ei.f fl (I - E,,).W which is invariant relative to A. By condition 6) we obtain a(A I Ea. fl (I - Eµ) ) C (- co, x] fl [µ, oo ) =0, i.e., Eat fl (I - E,,).Jf = (B), and therefore Ea = E,;E,,. Similarly E,, = ExE,. When µ 1 X we conclude that Ei = E,\E>, = E',E) = E) i.e., we have

proved 1)-6).

4 Spectral Topics and Some Applications

214

Now let c > 0. Since e

e

J -IIAII-0

v dE,,= JB'/2C

'

J-IIAII-0

v dFFBtn = JB"2C_,F- eBvz = JB1/2

PBiiz,

and IIAII

IIAII

v dE, _ -

f

e

v d(I - E,,) = - JB'/2Cf '

v d(I - FF)B'/z

e

e

= JB'nCC

IAII

I

f

I

IAll

P FB' 2 = JB1/2 CC 'CfI - FF)Bvz e

= JB'12(I- FF)Bin = A - JB'nFeBvz, it follows that 11

11

P dE, = A - JB'1z(Fo - F_o)B'/z -IIAII-0 We put S = JB'iz(Fo - F_o)B1/2. Since Fo - F_o is a projector on to Ker C, we have

Sz = JBvz(Fo - F-o)Bvz JB1 z(F0 - F-o)B1/2 = JBvz(F0 - F-o)C(Fo - F_o)Bin = 0. Similarly,

SE),=ExS=0 when X<0 and S(I-F>,)

=(I-E),)S=0 when X>0. Corollary 1.6: A bounded J-non-negative operator A has a J-spectral function E with a single critical point X = 0. Moreover,

a(E)\(0) = a+(E) U a_ (E) and a+(E) = (0, oo) fl a(E), a- (E) = (- oo, 0) fl a(E).

As E(A), when 0 = (a, l], a 0, 0 96 0, we put E(i) = Ea - Ea. The fact that this determines a J-spectral function of the operator A with a single critical point X = 0 can be seen from the properties of the function Ex investigated in Theorem 1.5. In particular, it follows from properties 2) and a+(E) = (0, oo) fl a(E), a(E)\(0) = a+ (E) U a_(E), and 3) that

a- (E) = (- oo, 0) fl a(E). Remark 1.7: It follows from condition 5) of Theorem 1.5 that if T is a bounded operator which commutes with A, then it also commutes with S.

Remark 1.8: Since (see the proof of property 5) in Theorem 1.5) the

§1 The spectral function

215

operators E, (X ;d 0) are the strong limits of certain polynomials of A, every subspace 2 which is invariant relative to A is invariant relative to Ea and S. Moreover, ' = E,,/ [ + ] (I - EE,)!, E 2' and (I - E,)2' are invariant relative to A, and a(A I E),?) C (- oo, X], and a(A I (1- Ea).T] C [X, co). If, moreover, ? is invariant relative to a bounded operator T which commutes with A, then E),2', (I - EE,)Y and S9? are also invariant relative to this operator.

We note also that by virtue of condition 3) in Theorem 1.5 the integral m p dE,. In future, where it is f II - o p dE. can be written in the form I% necessary to do so, the operators E, and S corresponding to the operator A II

will be denoted by the symbols EEA) and SA.

Corollary 1.9:

If A and D are commutating J-non-negative operators, then

E> A)Ea°) = Ea°)EX(A); SAES;°) = Ea°DSA = O when X < 0 and SA(I (I_ E)(°)SA = O when X > 0; SASD = SDSA = O; DSA = SAD = 0.

The permutability of the operators ExA), EaD), SA and SD follows from El condition 5) in Theorem 1.5 and Remark 1.7. In accordance with Exercise 2 on

2.§2, WsA is a neutral lineal. On the other hand, we conclude from the commutativity of the operators SA and EaD) when X 71 0 that it is invariant relative to EX(D). Consequently Ea°)JsA is a negative lineal when X < 0, and (I- EX(°")?sA is a positive lineal when X > 0 (see Theorem 1.5 condition 2)).

Hence it follows that E!°I is, _ (B) when X < 0 and (I_ E)(°)) 3 s, _ (0) when X > 0, i.e., Ex°DSA = 0 when X < 0 and (I - Ea°))SA = 0 when X > 0. By virtue of condition S in Theorem 1.5 the operators D and SA commute, and therefore the lineal ?sA is invariant relative to D. Since Y"sA C -,ip °, it follows from the inequality I [DSAx, Y]12 < [DSAx, SAx] [Dy, y] = 0 that DSA=0. It remains to verify the equality SASD = 0. Since

VE,(°) = s-lim (DEYD + D(I - Ef' )), we have flo

We now use the fact DSA = 0 and SD = D - f

p dEv°)SA = 0.

p

Let ./ = (A) be a commutative family of bounded J-selfadjoint operators. We shall say that it has the property 'J if every invariant dual pair of this family admits extension into a maximal dual pair invariant relative to .,d.

3

Later in Exercise 8

it

will be proved that a family of definitizable

J-selfadjoint operators has the property (i(11 if this family is either finite, or if for all the operator entering into it the `corners' P+ AP- I W- are completely continuous. But here now we shall prove the validity of the following result. Theorem 1.10: If d = (A) is a commutative family of bounded J-nonnegative operators, then d has the property 4) [l1 .

4 Spectral Topics and Some Applications

216

Let (2'+, 9-) be an invariant dual pair of the family .1. We have to prove +

that there are Y+ E Jf such that Y+ D Y+, Y+ [1] 2-, and l2' C Y+. We first verify this assertion for the case when the Y+ are definite. Let

f+(A)=C Lin((I-EaA))l a>0}, f_(A)=C Lin{EaAt.I X <0}, Lin(f+(A)IAE.1), and 'V=C Lin(91s.,IAEd) It follows from Corollary 1.9 that f + (,d) [1] 6'_ (. /) and /V [1] 41 + (.d), and from

Remark 1.8 that '+ C (f + (,t))[1[. By virtue of this same Remark 1.8, SAY± C Y+ (A E V). But since the '+ are definite and MA, C

°, we have

SAY+ = (B} (A E .z), and therefore q+ [1] .u{

From what has

We consider the subspaces ?'+ = C Lin (Y+, f +

been proved above it follows that 2+ [1] 2_. We verify the inclusions Y+' C e + . In accordance with Exercise 11 on II. §3 the subspaces +

that .4 "[1] C Lin(99+,f+(,RI)}. Thus (Y', Y-' ) is an invariant dual pair of the family d and it contains (2'+, Y_). We Let (2+, 2-) be an arbitrary maximal dual pair containing + By construction for all A E , '. shall prove that Al C Y± )+ C Y, 1 [1 C 6 _ (A) [1[ and therefore AY+ C A6'-(A)1'1. Since C

Lin (Y+, f +.j7)) C JP

A = SA + f

.

It

remains

to

use

the

fact

v dE(PA), we have

Ae_(A)[1[ C C Lin(J1s,,,6+(A)}C Y+ C Y+. Consequently, AY+ C Y+. Similarly AY- C 2- (A E d). Thus the theorem is proved when 2'+ are definite. Let (9+, Y_ ) be an arbitrary dual pair invariant relative to .,l, and let VY)=C Lin(Y+,Y+ n.9°}. Then (Y(+'), YIP) is an extension of the pair (Y'+, Y_), and Y1+) [1] 241), Ac+) n A(l) = Y4) n Y(+)[ll = Y,() n i W[L] _ Yo,

and dYT C YQI). We pass to the consideration of the factor-space e= Yd1[14, and of the family of Jnon-negative operators It = (A} induced in it by the family e1, where I' 1) = YTIY0. In and its invariant dual definite pair accordance with what has been proved above, the family . ,/ has an invariant

maximal dual pair (+, 7-) containing (.+), j'T). In accordance with Exercise 7 on L§5 the subspaces 7 + = (x + E f c± I X± E dual pair in .,Y, and C Y+ D Y+.

i +) form a maximal

4 In conclusion we show how the results in §1.2 can be applied to obtain a (JI, J2)-polar decomposition of a (J1, J2)-bi-non-expansive operator. But first we give a definition.

Definition 1.11: A bounded J1-selfadjoint operator R is called a J1-module of a bounded operator T acting from a J1-space .01 into a J2-space . 2 if c(R) C [0, oo), R2 = T'T and Ker R = Ker T`T. Theorem 1.12: Every (J1, I2)-bi-non-expansive operator V has at least one J1-module R and admits a `(J1, J2) -polar decomposition' V = WR, where W

is a (J1, J2)-bi-non-expansive operator which is simultaneously a (J1, J2)semi-unitary operator or is conjugate to a (J1, J2)-semi-unitary operator.

§1 The spectral junction

217

E The operator A = I, - V` V is a bounded Ji-non-negative operator. Since from 2.Theorem 4.30 and 2.Remark 4.29 we have a(V` V) C [0, co), it follows

that a(A) C

1). Let Do = (Z, 1], and let E(go) be the corresponding value of the J1-spectral function of the operator A. From the definition of the J1-spectral function it follows that E(Do), 1 and (I, - E(Do)), 1 are invariant

relative to the operator V` V, and moreover a(V`VI E(Do).1) C [02] and a(V`V (I, - E(Do)).Y,) C [Z, 11 V`V 11]. In accordance with Corollary 1.6 the subspace E(Do)1 is uniformly positive. Since V` VI E(Do)W1 is an operator

which is selfadjoint relative to the form [ , ], I E(Do)1 and has a non-negative spectrum, it is non-negative and therefore it has a non-negative square root R1. Let F be a Jordan contour, enclosing the interval [Z, 11 V` V111, symmetric relative to IR and not containing X = 0 within itself, and let ,X be an

analytic function taking positive values for positive X. Then (see [XXII]) the

operator R2 = -(I/2iri)f r;X(V`VI (I, - E(Do)). 1 - XI1) ' dX is bounded,

= V`Vj(I, - E(Do)),J'1, a(R2) C [1/,2,jj V`V11], and, because IF is symmetric relative to R, the operator R2 will be J1-Hermitian. Since R2

E(Do).W1 [ + ] (I, - E(Do))Y1 = MI, the operator R such that Rx = Rix if x E E(Do)J'1 and Rx = R2x if x E (Il - E(Ao))J"1 is J1-selfajoint and bounded, R2 = V` V, Ker R = Ker V` V, a(R) C [0, oo), i.e., R is the J1module of the operator V. Vi = V I E(Do)4e1 and We now consider the operators is V2 = V I (I, - E(Do)) 1. Since E(Do)de, uniformly positive, (I, - ElDo)).1 contains the uniformly negative subspace which is maximal in

.i'1, and therefore V(I1 - E(Ao))., contains the uniformly negative subspace maximal in W2. From 1. Corollary 8.14 we conclude that .?Iv, (= v,) is projectionally complete, and MW is uniformly positive. Since E(Do).k1 and (I, - E(Do)).1 are invariant relative to V`V it follows that V` V,l C E(Do) 1, V` v, C (Ii - E(Do)), 1, and ,v, C XV,1. The operator V, as an operator acting from the Hilbert space E(Do)W1 with the scalar product [ , ] 1 into the Hilbert space 91IZl with the scalar product ]2 has the decomposition V1 = W1R1, where W1:E(Ao),,Y1-,W1`1 is [ semi-unitary or is adjoint to a semi-unitary operator. It is now sufficient to put Wx= W,x if XE E(Do)Mi, and Wx= V2R2- 'x if xE (Il - E(Do)). 1, and we obtain that V = WR and W satisfies the requirements of the theorem.

Exercises and problems 1

2

Prove that J-selfadjoint operators A of the class K(H) with a real spectrum have a unique J-spectral function with a finite number of critical points I X, , where X, are the eigenvalues of the operator A with a degenerate Ker(A - ail) (Azizov). Hint: Use the result of Exercise 15 on 3.§5 and the decompositon analogous to 3.(5.3) for a bounded J-selfadjoint operator of the class K(H). Similarly to Definition 1.3 introduce the concept of an eigen J-spectral function of a

J-unitary operator and prove that J-unitary operators UE K(H) have a

J-spectral function, which is, moreover, unique (Azizov).

4 Spectral Topics and Some Applications

218 3

Let Il, be a space with x negative (respectively, positive) squares, let 9' be a x-dimensional non-positive (respectively, non-negative) invariant subspace of a a-selfadjoint operator A, and let k,, X2.... , X, be the eigenvalues (taking multiplicity into account) of the operator A 17. Prove that the polynomial p(X) = II k= ,(k - X) (respectively, - p(k)) definitizes the operator A (I. Iokhvidov and M. Krein [XV] ).

4

Prove that if A is a definitizable J-selfadjoint operator, and (X,, are all the non-real roots of the definitizing polynomial p, and (k;, a;) C a(A), then a(A)\IR = (k;, X;); ,

IX,, a;] C ap(A),

Yx,(A)

and

't,;(A)

are

subspaces

(i = 1, 2, ... , k), and k

.,_ [+] (.(A) +u1'X;(A))

"Y"

i=1

and A :.JY' -.,Y' is an operator definitizable in " Wwith a real spectrum (Langer [4], [5]). 5

Prove that a definitizable J-selfadjoint operator with a real spectrum has a function which is, moreover, J-spectral E unique. Further, a(E) = a+(E) U a_ (E) U s(E), and when A fl s(E) = 0 and the A are finite the operators E(0) are the strong limits of polynomials of A (Langer [4], [5] ).

6

Prove that operators A of the class K(H) or definitizable operators are bounded if and only if a(A) is a bounded set (Azizov; Langer [4], [5]).

Hint: Use the decomposition 3.(5.2) and the results of Exercises 1, 4, and 5 respectively. 7

Prove that if A E K(H) or if A is a definitizable operator, and if a(A) _ (X;)1 , then .,Y = Yx,(A) + 99,,Z(A) + ... + 99ak(A), and the lengths of the Jordan chains of the

operators A I ',,;(A) are bounded (Azizov; Langer [4], [5]). Hint: Use the expansion 3.(5.4) and the results of Exercises 1, 4, 5. 8

Prove that if d is a commutative family of bounded definitizable J-selfadjoint operators and either .dd is finite or the `corners' P+AP- I . E .v'm for all A E d, then d has the property X111 (Langer [9], [13]).

9

Prove that if A is a definitizable J-selfadjoint operator, and if D is a finitedimensional J-selfadjoint operator, then A + D is a definitizable operator (Jonas and Langer [1]).

10

Prove that a J-selfadjoint operator of the class K(H) is definitizable if and only if function Exercise is that its E (see 1) such J-spectral a(E) = a+ (E) U a_ (E) U s(E) (Azizov).

Hint: Use Exercise 15 on 3.§ and the results of Exercises 1 and 5 taking into account a decomposition similar to 3.(5.2) for a bounded J-selfadjoint operator of the class K(H). 11

Prove that the J-spectral function of a J-selfadjoint operator A E K(H) with a(A) C R is regular if and only if all the 99),(A) are non-degenerate (Azizov). Hint: Use the expansion 3.(5.4).

12

Prove that if a J-selfadjoint operator A satisfies the condition [Ax, x] # 0 when

[x, x] = 0, then there is a constant a such that either A - aI or aI - A is a J-non-negative operator with a definite kernel and A = aI + J°°m(µ - a) dE (Personen [1], M. Krein and Shmul'yan [3]). 13

Let T be a strict plus-operator acting from a J,-space .JY, into a Jz-space .IY2, and

§2 Completeness and basicity of root vectors of J-dissipative operators 219 let a(T`T) > 0. Then the operator T has a J,-module and only one (cf. M. Krein and Shmul'yan [3] ). Hint: See the proof of Theorem 1.14. 14

Prove that a J-selfadjoint operator A is definitizable if and only if there is a function f, holomorphic in the neighbourhood of its spectrum and of the point X = oo, such that f(A) is a non-negative operator (Langer [4], [5]).

15

Prove that if A is a J-selfadjoint operator of the class K(H), then there is a decomposition Ye = Y I [+].YY2, where .', C 9-A, A.Yk'i C X j (i = 1, 2), a(A I Wi) C [m, M], - oo < m < M < oo, and A I X2 is similar to a selfadjoint operator and a(A I .YP2) C (- oo, m] U [M, co) 1. Iokhvidov [11). Hint: Use the results of Exercise 1.

16

(Azizov,

Gordienko

and

Let (U,)mm be a one-parameter group of the class (Co), (i.e., U,+S = UU5, Uo = I, and U,x U,ox when t to, and for all x E .Y') consisting of J-unitary operators,

and let iA be an arbitrary operator of this group. Prove that (A` _) A E K(H) if and only if (U,)__m E K(H) and then U, = exp(itA) (Azizov, Gordienko and 1. Iokhvidov [1]). Hint: Use the results of Exercises 1, 2, 15 and also Naymark's article [3].

§2.

Completeness and basicity of a system of root vectors of J-dissipative operators

In this section, if nothing to the contrary is stated, all the operators are continuous and are defined on the whole space. The spaces are assumed, generally speaking, to be non-separable. Our references to the monograph 1

[XI] in which the spaces are assumed to be separable are valid in this respect, that either separability is not essential for the validity of the actual results or our operators here are acting in a separable space. We recall that a system of vectors (e«) is said to be complete in a space .W if .Y' = C Lin (e,.. Everywhere it is to be understood that a E A, where A is a certain set of indices. fa) = 6(a, 0) A basis (f.) of a space Y is said to be orthonormalized if

(a, 0 E A). If T is a bounded and boundedly invertible operator, then the system (Tf') will also be a basis in YY and it is called a Riesz basis. Here we shall investigate the questions of the completeness and basicity of the roof vectors (i.e., the eigenvectors and principal vectors) of J-dissipative,

and in particular, of J-selfadjoint operators of the class K(H) and of J-selfadjoint definitizable operators. We introduce the notation (I (A) = C Lin(2'x,(A) I X E ap(A)), fo(A) = C Lin(Ker(A - XI) I X E ap(A)).

(2.1) (2.2)

By definition fo(A) C ('(A) and therefore dim. %"/ (A) S dim .Yt°/eo(A). We shall first investigate the given questions in the case of J-dissipative operators of the K(H) class. We note that if A E K(H) and if X = X E ap(A), then the root lineal 27 (A) is closed and its non-degeneracy is equivalent to its

220

4 Spectral Topics and Some Applications

projectional completeness (Exercise 5 on 3.§5). Since A E K(H), there is not more than a finite number of real points X to which correpond degenerate kernels Ker(A - X T) of this operator. We shall call the set of such points, as also in the case of J-selfadjoint operators of the class K(H) (see Exercise 1 on § 1), the set of critical points and denote it by the symbol s(A ). Lemma 2.1: Let A be a bounded J-dissipative operator of the class K(H) whose non-real spectrum consists of normal points. Then non-degeneracy of f(A) and non-degeneracy of Lin (Y), (A) I X E s(A) } are equivalent. Moreover, if f (A) is non-degenerate, then it is projectionally complete. Since (see Exercise 5 on 3.§5).,,(A) when X = A can be expressed in the

form of a sum of subspaces invariant relative to A :.,,(A) = ' [+] alt:,, where dim IV,, < co, and #), is a projectionally complete subspace and '46' C Ker(A - XI), we conclude from 2.Corollaries 2.16 and 2.18 that Lin(2',,(A)I X E s(A)) is a subspace and its non-degeneracy is equivalent to its projectional completeness. Now let Lin {27), (A) I X E s(A)) be degenerate, and let 2' be its isotropic part. Then ..2 C Lin{.il'), X E s(A)), and therefore q is finite-dimensional. By virtue

of 2.Lemma 2.19 we have AY C Y. Consequently there is in Y at least one eigenvector xo of the operator A. By its choice xo [1] Lin[114,(A)I XEs(A)), and by virtue of 2.Corollaries 2.16 and 2.18 we have xo [1] 1,, (A) when µ ¢ s(A), i.e., xo j1] 6(A). Since xo E f (A), we see that degeneracy of Lin (2'x(A )l X E s(A)) implies the degeneracy of 6(A). Conversely, let ,"(A) be degenerate, and let Y, be its isotropic part. Then AY, C 2,, by virtue of 2.Lemma 2.19. We first assume, and later prove, that dim Y, < oo. If it is, then there is in 21 an eigenvector yo of the operator A : Ayo = Xoyo From 2.Corollary 2.25 and the definition of the set of critical points we conclude that X0 E s(A), and therefore Lin(2,,(A )l X E s(A)) is a degenerate subspace. We now verify that ., is finite-dimensional, and at the same time that if f (A) is non-degenerate, then it is projectionally complete. To do this it suffices to show that there is in 44'(A) a projectionally complete subspace .4" such that dim e (A) I ./V< oo. Since A E K(H), by virtue of 3.Corollary 5.18 there are in every subspace Ker(A - XI) when X = X semi-definite invariance subspaces 9a , maximal (in Ker (A - XI)), of the operator Vo. Then r+ = C Lin(1,,(A), 2,; I Im it > 0, X = X} is a subspace completely invariant relative to Vo and invariant relative to A, and Y1- = C Lin (1,, (A ), 2- I Im µ < 0, X = X } is a subspace invariant relative to Vo

and A. By virture of 2.Corollaries 2.18 and 2.22 it follows that the first of these subspaces is non-negative and the second is non-positive. Taking

3.Corollary 5.4 into account this implies the inclusions Y'+ E h'- . Let 190 = 2,+ fl Y_ . Then Y+ + 2'_ = 20 [+] .'I where .'U is a projectionally complete subspace. It remains to notice that dim "(A) .Y'+ + 2_ < co, and therefore if f (A) is degenerate, then its isotropic part is finite-dimensional, and if it is non-degenerate, then it is projectionally complete.

§2 Completeness and basicity of root vectors of J-dissipative operators 221 Remark 2.2: The condition in Lemma 2.1 about the normality of the non-real points of the spectrum of the operator A is used only in the proof of the fact that degeneracy of f(A) implies the degeneracy of Lin (9'),(A )I X E s(A)I. Therefore the remaining assertions are valid even

without this condition, and in particular, if e (A) is non-degenerate and therefore projectionally complete, then the operator Vo I 6,'(A) belongs to the class H. In fact, the subspace Y+ constructed in the proof of Lemma 2.1 has in ,,(A) the property def 9?+ < oo. From Exercise 9 on 3.§3 we conclude that Vo I 6'(A) is a bi-non-contractive operator relative to the form [ , ] I 6,(A) and it has a completely invariant subspace JO+ E .itl+ (f (A)). In combination

with 3.Theorem5.2 and the fact that Vo E H, this is sufficient to prove the validity of our assertion. Moreover, from Exercise 17 on 3.§5 it follows that if P is the J-orthoprojector on to f (A) then the operator PVo I f (A) H and it is P*JP-bi-non-contractive.

Corollary 2.3: Let A E K(H) be a completely continuous J-dissipative operator. Then non-degenerary off (A) and 'o(A) are equivalent. The operator A satisfies the conditions of Lemma 2.1. Moreover, since all X ;4 0 are normal points of the operator A, it follows by virtue of 2.Corollary 2.25 that all the T,,(A) when X = X # 0 are non-degenerate. Consequently, degeneracy of 2o(A) and degeneracy of Lin(9?x(A)I X E s(A)) are equivalent. It remains to use Lemma 2.1.

2 The following two lemmas are key results in solving questions about the completeness of systems of root vectors of J-dissipative operators.

Lemma 2.4:

Let the operator satisfy the conditions of Lemma 2.1, let f(A) be non-degenerate, and let P be the J-orthoprojector on to Then A, = PA I (A)[) E K(H) and ap(A,) = 0. That the operator A, belongs to the class K (H) follows from the fact that if A E K(H, Vo), then by virtue of Exercise 17 on 3.§5 PVoI f,`(A)'1] E H and therefore A, E K(H, PVo I f(A)I1)Now

let X= X E ap(A). Since e '(A)1'1 is a P*JP-space, and A, is a P*JP-dissipative operator, it follows from Xo = Xo that Xo E ap(C`). Let Aixo = Xoxo. By virtue of 2.Corollary 2.17 A,xo = Xoxo. On the other hand, Al = A` I e '(A)1`1, and again by virtue of 2.Corollary 2.17 Axo = Xoxo; but this contradicts the fact that xo [1] f (A) and f(A) is non-degenerate, and therefore ap(A,) fl R = 0.

Now suppose that o # Xo E ap(A, ). By virtue of 2.Theorem 1.16 Xo E ap(Al) U a,(Al). We again use the fact that A i = A` I f (A)[1). Since by

hypothesis all the non-real points of the spectrum of the operator A (and therefore, by virtue of 2.Theorem 1.16, also of A`) are normal eigenvalues, it

follows that oEap(A`), which implies in the present case the inclusion

4 Spectral Topics and Some Applications

222

Xo E kp(A), i.e., Xo is a normal eigenvalue of the operator A and therefore also

of the operator A I "(A). Consequently, there is a decomposition f (A) = 2>(A) + such that Af' C f' and Xo E p(A I e,"). Then follows from Xo E p(A I f') that Xo E apl A2), where A2 = QA I (f' [+] f (A)I`), and Q is the

.W = Y.

.(A) + (f' [+] f (A)WWW ).

Since

Xo E ap(A1)

it

projector on to 6," [+] f (A) 1' parallel to 27x0 (A ). Let A2xo = koxo. Then (A - XoI)xo E 27>(A ), i.e., xo is a root-vector of the operator A corresponding to X0 and not lying in 27ao (A)-we have obtained a contradiction Let Jl be a certain complex of properties invariant relative to a bounded projection and an equivalent renormalization of the space, and let each bounded dissipative operator having this complex of properties have a complete system of root vectors. Then every J-dissipative operator A E K(H) with a non-degenerate f (A) also has a complete system of root vectors in M. Lemma 2.5:

We suppose the opposite. Since the properties in the complex .W are invariant relative to a bounded projection, they are in particular invariant also relative to a J-orthogonal projection. It follows from Lemma 2.4 that either

Lemma 2.5 is true, or there is a bounded J-dissipative operator A E K(H) having this complex of properties and such that ap(A) = 0. We suppose the latter holds. From 3.Theorem 5.1b we have that the operator A has at least ± ± one pair of invariant subspaces 27± E . i fl h . Since ap(A) = 0, the Y± are uniformly definite subspaces, and therefore .e = 2+ + 27-. Since the operators ±A I Y± have the properties of the complex ,* and are dissipative in relation to the scalar product ± [ , ] 127± which is equivalent to the original one, it follows by the hypothesis of the theorem that

f (A I 2±) _ 2'- and therefore up(A) - 0-we have obtained a contradiction.

Lemma 2.5 enables us to transfer to the case of bounded J-dissipative operators of the class K(H) a whole series of assertions about the completeness of the system of root vectors for ordinary dissipative operators since, as a rule,

the conditions in these assertions are invariant relative to the operations of bounded projection and equivalent renormalization of the space (see, e.g., [XI] ). We shall not cite all these assertions in the main text, but we shall introduce some of them later in the form of exercises. We show by the example of Theorem 2.6 how to carry out these exercises. We also prove Theorem 2.8;

this is interesting because it cannot be generalized, not even in the case of operators of the class H (see Example 3.3 below). But first we recall some familiar terms and introduce some new ones. .02 be a completely continuous operator, and let [s, (A )J be Let A :.W'1 the set of eigenvalues of the operator (A *A ) 1/2 (taking multiplicity into account), or in other words, [ s. (A )] is the s-number of the operator A ([XI]). We say that A E .y'p if Es. (A) < oo. In particular, if p = 1, then A is called

a nuclear operator, and if p = 2, A Hilbert-Schmidt operator. We remark that even if .,Y, is non-separable, all the s-numbers except perhaps for a count-

§2 Completeness and basicity of root vectors of J-dissipative operators 223

able set are equal to zero. It will therefore be convenient to us to suppose below that sk (A) * 0 when k = 1, 2, ... , v; v < oo. Usually the s-numbers

sk(A) are numbered in the order `biggest first', and then the formula min (II AK 11

that if equivalent

1 dim K < n) holds ([XI]). Hence, it follows in particular

are the s-numbers of the operator A in another norm

there is an m > 0 such that (n = 1, 2, ...). Let lea) be an orthonormalized basis, and { fa } be a J-orthonormalized Riesz basis of a J-space W. The number sp A = Ea(Aea, ea), where A E .9'1,

to the

first

one, then

is called the trace of the operator A. As is well-known (see, e.g.,

[XI] )

sp A = paXa, where (Xa) is the set of eigenvalues of the operator A taking multiplicity into account, and therefore sp A does not depend on the choice of the orthonormalized basis and the equivalent renormalization of the space. This enables us to write the formula for the trace in the equivalent form: sp A = Z. [Afa, fa] sign [fa, fa]. All three of these formulae will be used later.

Let A E K(H) be a completely continuous J-dissipative Theorem 2.6: with a nuclear J-imaginary component A,, and let operator

n (p; AR)Ip = 0, where AR is the J-real part of the operator A, (n = 1,,2.... ) n(p;AR) is the number of numbers of the form situated in the segment [0,p]. Then e(A)=,W if and only if Yo(A) is limp

non-degenerate.

The necessity for non-degeneracy follows from Corollary 2.2. Sufficiency: From Corollary 2.2 we obtain that if !Fo(A) is non-degenerate,

then f(A) is also non-degenerate. Let P be the J-orthoprojector on to d(A)HHH, and let A, = PA I f (A)1`1. Since (see, e.g., [XI], 2.2.1), we have limp-. n(p; (A,)R)/p = 0. Therefore if f (A) * .,Y, 11

we can by virtue of Lemma 2.4 suppose without loss of generality that the original operator A is a Volterra (i.e., A E Y. and a(A) = {0)) J-dissipative operator of the class K (H) with 0 ¢ up (A) and we shall prove that the equality limp - . n (p; AR)p = 0 is impossible. Indeed, it follows from 3.Theorem 5.16 that the operator A has invariant subspaces Y ± E, it ± fl h ± . They are

non-degenerate; for otherwise it would follow from 2.Lemma 2.19 that ap(A) 3;60. Consequently the Y' are uniformly definite and therefore .N'= M'+ + Yl-. Suppose, for example, that 2+ *- (0). We consider the It is a Volterra dissipative operator relative to the operator A+ = A (definite) scalar product [ , ] +, and limp-. n(p;AR )gyp = 0. Since the scalar product [ , ] I Y'+ is equivalent on 11 '+ to the original one, it 19+.

I

follows that limo-. n(p; AR )/p = 0, where fi(p,AR) is the number of numbers of the form are the s-numbers in the interval [0, p], and the of the operator AR relative to the scalar product [ , ] 191+. Let n+ (p; AR) be the number of numbers of the form 1/X (AR) in the interval [0, p], where the

lXn (AR)) are the positive eigenvalues of the selfadjoint operator A. Since

4 Spectral Topics and Some Applications

224

0 < n+ (p; AR) < n (p; AR +), we have limp - m n+ (p; AR )Ip = 0. It follows from

[XI] 4.7.2 in this case that limp-.. n+ (p; AR )Ip = (1 fir) sp Ai . Consequently

sp A; = 0. Since Al' is a non-negative operator, sp Al = 0 implies At = 0, i.e., A = AR is a completely continuous self-adjoint operator, and therefore ap(A+) 0-we have obtained a contradiction. We arrive similary at a contradiction if Y- ;4 (B). Later we shall more than once make use of the following simple proposition.

If A is a selfadjoint operator with a spectrum having no more than a countable set of points of condensation, then fo(A) =, and Lemma 2.7:

is an orthonormalized basis composed of the eigenvectors of the in Wthere ' operator A. Since Y) ,(A) = Ker(A - XI) and Ker(A - XI) 1 Kerr(A - µl) when X ; µ, there is in fo(A) an orthonormalized basis composed of eigenvectors of the operator A. It remains to verify that fo(A)1 = 101. Suppose this is not so.

Since a(A I fo(A)1) contains no eigenvalues, and the set of points of condensation of a(A I 6,,o(A)1) can be no more than countable, there is at least

one isolated point of the spectrum of the operator A(fo(A)1. This point must, as is well-known (see, e.g., [1]), be an eigenvalue-we obtain a contradiction.

A J-dissipative operator A will be called simple if there is no subspace invariant relative to A and A` on which these operators coincide. Theorem 2.8: If an operator A with a nuclear J-imaginary component Ai is a simple 7r-dissipative operator or if a(A) has no more than a countable set

of points of condensation, then f (A) = II, if and only if the lineal Lin(3),(A)l X E s(A)) is non-degenerate and Eq Im X. = sp Ar, where X. traverses the set of all eigenvalues of the operator A taking multiplicity into account.

Since A = AR + iA1 and the non-real spectrum of the ir-selfadjoint operator AR consists of normal eigenvalues, it follows from Ar E .9', that the non-real spectrum of the operator A consists of normal eigenvalues. Therefore

all the points of condensation of a(A) lie in R. Moreover, by virtue of 2.Theorem 2.26, a(A) _ . { "a [+] lla, where A.;{ ",, C .'{ x, dim . /I'x < oo, /la is non-degenerate, and therefore and //x C Ker(A - XI) and Lin(91>,(A)I XEs(A)) is a subspace.

Suppose that f (A)=H,. We then obtain from Lemma 2.1 that Lin ('),(A )l X E s(A )) is non-degenerate. We consider the subspace 9'= C Lin (..//a I X E a(A) fl IR). This is a non-degenerate subspace: for other-

wise it would follow from 2. Lemma 2.19 that the operator A I U' has an eigenvector xo ;4 0 isotropic in U', and by virtue of the fact that 9' is 7r-orthogonal to C Lin(. a, U',,(A)j X = X,µ * µ) it would follow that xo [l] f (A)-we obtain a contradiction. We consider the decomposition

§2 Completeness and basicity of root vectors of J-dissipative operators 225

fl = Y[+] o[l]. Since AY C M and A`Y C !, it follows that A11 C 'Ill and the operator A I -1' satisfies the same conditions as A but with this difference that all the root subspaces of the operator A I YI' are finitedimensional. Since Y C "(A) we shall suppose without loss of generality that ' = (B). Since (by construction) ./Vx Rd (B) only when X E s(A ), so aa(A) fl IR

has not more than a finite number of points, and therefore the operator has not more than a countable set of eigenvalues. Consequently, just as in [XI],

1.4.1, an orthonormalized basis (ek) can be constructed in H. such that (Aek, ek) = Xk and Xk runs through the set of eigenvalues of the operator A taking multiplicity into account. Hence, [(I/2i)(A - A*)ek, ek] = Im Xk. Since

A,E.q'1, and A,-(1/2i)(A-A*)=(1/2i)(-A`+A*)=(1/2i)(- JA*J+A*) is a finite-dimensional operator, and moreover

sp(I (JA*J-A*)I sp(21 J(A*-JA*J)J) =sp(I _

(A*-JA*J))

-sp2 (JA*J-A*)l

it follows that sp

(f (JA *J - A *) I = 0, 21 (A - A *) E J,

and sp A, = sp (2i

(A -A*)

Conversely, let Lin (Yx(A )I X E s(A )) be non-degenerate and let sp A, = Z. Im Xa. Again by virtue of Lemma 2.1.E (A) is non-degenerate. Again without loss of generality we suppose that 99 = C Lin(,/tt,, I X _ ) = (B). We con-

sider the operators A' = A E(A) and A, = PA I E(A)where P is the

7r-orthoprojector on to E (A) [lI . Since the operator A' satisfies the same conditions as the operator A, we have sp A,'= Ea Im Xa. Let (fk')) and (ffz)) be wr-orthonormalized based in 6(A) and E (A )Ill respectively. Then

sp A, = Z [A,fk'), fk')]sign [fk'), fk')] + k

= Lj k

[A,f«z),

f.(2) ]

sign [f.(2), f.(2)]

a

[Alfk'),

Im Xk +

fkl)]slgn[fkl), fk')] +

a

[(A 1),f(2), f.(2)] sign

[(A1)lfaz), 1«z)]sign

[f«z), fz)l

[f(2),1(2)]

k

which, by virtue of the equality sp A, = Ek IM Xk implies the equality [(A, )rf«z), f«z)] sign [f«z), f«z)] = 0.

Suppose for definiteness that H. is a Pontryagin space with x negative squares. Then f(A) is also a Pontryagin space with x negative squares, and therefore

4 Spectral Topics and Some Applications

226

f (A )t1J is a positive subspace. Hence it follows from Z [(At )rff2) f .M] sign

0

a

that (A,), = 0, i.e., A, = Ac. Since Ai = Ac I f (A)111, we conclude from 2. Theorem 2.15 that Af(A)111 Cf(A)[1] and AI 6(A)111 =A`I 6,(A)111 = A,. If A is a simple T-dissipative operator, it follows from this that f (A)111 = (0). If, however A is not simple but a(A) has no more than a countable set of points of condensation, then the operator A,, which is self-adjoint relative to the definite scalar product [ , ] I 6(A)[11, also has this property. Therefore, by virtue of Lemma 2.7, ap(A,) 0 if r(A)111 (0); f(A)111 = (0).

but,

by Lemma 2.1, ap(A) = 0

which

implies

that

Corollary 2.9: If A E K(H) is a nuclear J-dissipative operator, then nondegeneracy of 20 (A) is equivalent to the equality "(A) =,_W.

It follows from Corollary 2.2 that if 6(A)=, , then 2o(A) is nondegenerate. Conversely, let Yo (A) be non-degenerate. Then again from Corollary 2.2 we

obtain that 6,(A) is also non-degenerate. We make use of Lemma 2.1, by virtue of which it suffices to prove that when Y ;4 (0) there are no Volterra nuclear J-dissipative operators A of the class K(H) with 0 I< ap(A ). Let us suppose the contrary. As in the proof of Theorem 2.6 we conclude from ap(A) = 0 that the operator A has a maximal uniformly definite invariant subspace 2. Suppose for definiteness that 2 E -it' (the case 2' E ,fl- is verified

similarly). We consider the operator A I Y. This is a nuclear dissipative operator relative to the form [ , ] I Y. Since sp(A I 2') = EaXa(A I 2'), we have sp(A 12)r= Ea Im X.(A I 2). By Theorem 2.8 ap(A I .') ;4 0-and we have a contradiction. 3 Before we present results about the basicity of systems of root vectors of J-selfadjoint operators, we introduce.

Definition 2.10: A basis (fa) of a J-space 0 is said to be almost J-orthonormalized if it can be presented as the union of a finite subset of vectors and a J-orthonormalized subset, these subsets being J-orthogonal to one another. Definition 2.11: A basis (fa) is called a p-basis if there is an orthonormalized basis (ea) and an operator T E .f', such that fa = (I + T)ea (a E A ). Theorem 2.12:

Let A be a continuous J-selfadjoint operator of the class

K(H), and let a(A) have no more than a countable set of points of

§2 Completeness and basicity of root vectors of J-dissipative operators 227 condensation. Then:

a) dim J '/ (A) 5 dim , lfo(A) < oo; b) fo(A) =. if and only if s(A) = 0 and Y),(A) = Ker(A - XI) when X

X;

c) f(A)= Jt if and only if Lin(Yx(A)I X E s(A)) is a non-degenerate subspace;

d) if fo(A) =,-W (respectively, 6(A) = M), then there is in .0 an almost J-orthonormalized Riesz basis composed of eigenvectors (respectively, root vectors) of the operator A; e) if --o (A) = W, then there is in' a J-orthonormalized basis composed of eigenvectors of the operator A if and only if a(A) C FR; f) the bases mentioned above can be chosen as p-bases if and only if the

operator A has an invariant subspace Y+ E J(

with an angular

operator Ky,, E 21p.

a) Since fo(A) C f(A) it follows that dim . lf(A) < dim ,lfo(A). We use 3.Corollary 5.21 and an analogue of 3.Proposition 5.14 for J-selfadjoint operators of the class K(H) and we obtain that dim f(A)/fo(A) < oo. Consequently to prove the inequality MI-6(A) < oo. dim Yelfo(A) < oo it suffices to show that dim

Let 21+ E u+ fl h+ be an invariant subspace, existing by virtue of 3.Theorem 5.16, of the operator A, let _T- = 21_ 1I, and 21o = 2+ n 21-. Since ot'l = Lin (21+, 2-) and dim , 21,' = dim 20 < oo, to prove the inequality under discussion it is sufficient to establish that 201, = .9(A 121 ), or, equivalently, that the f space ie = XolI-To coincides with f (A ), where A is the J-selfadjoint operator generated by the operator A. But the latter follows from the fact that ie = C+ [ + ] 21 21 ± I21o are uniformly definite subspaces

invariant relative to A, and the operators A/40± satisfy the conditions of Lemma 2.7. b) Let fo(A) = M. It follows immediately from 3.Formula (5.4), from the

definition of the set s(A) and the fact that 14(A) is J-orthogonal to 2',(A) when X # µ, that s(A) = 0 and 2),(A) = Ker(A - XI) when X ;d X. Conversely, let s(A) = 0 and 2),(A) = Ker(A - XI) when X ;4 X. We again

use 3.Formula (5.4) and without loss of generality we shall suppose that a(A) C R. Since s(A) = 0, all the kernels Ker(A - XI) are non-degenerate, and therefore 2a (A) = Ker(A - XI). Hence, by virtue of Lemma 2.1, ','(A) (= fo(A)) is also non-degenerate. But since dim fo(A)WWW = dim ,/fo(A) < oo and Afo(A)11J C fo(A)111, we have fo(A)[L] = (B}, i.e..W = fo(A). c) If f (A) = ', then 6,(A) is non-degenerate, and by Lemma 2.1 Lin (2a(A) I X E s(A)) is a non-degenerate subspace. Conversely, let Lin(2),(A) I X E s(A)) be a non-degenerate subspace. Again

by virtue of Lemma 2.1 6(A) is non-degenerate and ' = 6(A) [+] f (A)1, moreover Af(A)WWW C f(A)1-, AI f(A)t1 E K(H), and ap(AI f(A)WW) =0.

228

4 Spectral Topics and Some Applications

Hence it follows that f (A )f11 = L + [+] _ , where .+ are maximal (in ('(A)1'1) uniformly definite subspaces invariant relative to A. Since a(A 12"±) C a(A), the sets a(A I Y±) have no more than a countable set of points of condensation. By Lemma 2.7 aa(A I Y+ ) = 0-we obtain a contraction with the fact that ((av(A) 12+) U ap(A I Y=) = (av(A)E(A)111) = 0.

d), e). First of all we note that if an operator A E K (H) has non-real eigenvalues, then, by virtue of the neutrality of the eigenvectors corresponding

to them and 3.Formula (5.4), there is in .W no J-orthonormalized basis composed of eigenvectors of the operator A. Now let f (a) = .1 ', and , (A) _ "a [+] ll,

when X E s(A), where A4' ,, C ,",,, dim 4'X < co, ,(4 C Ker(A - XI) and ill,, is projectionally complete (see Exercise 5 on 3.§5). Since s(A) consists of a finite number of points, and by virtue of Lemma 2.1 all the 2,,(A) are non-degenerate when X E s(A ), so, taking 3.(5.4) into account, M = '1 [+] W2, where , and W2 are invariant relative to A, and Ye, = Lin (Y,, (A ), ail "a 114 ;d µ, X E s(A)J. Consequently dim Wi < oo, and a(A 1.02) C IR1 and moreover fo(A I. 2) = f(A I Y2) =.3f2. By virtue of Exercise 15 on 3.§5 the operator A I W2 E K(H). Therefore assertions d) and e) of the theorem will be proved if we show that the conditions a(A) C 91 and fo(A) =.e imply the existence in

.w' of a J-orthonormalized Riesz basis composed of eigenvectors of the operator A. We verify this.

Let (Y+, Y-) be a maximal dual pair invariant relative to A, and and 2a = Ker(A - XI) n Y±. It can be verified, just ,

let Y± E h ±

in the proof of assertion a), that fo(A I Y+) = 2'±. Since Ker(A - XI) [ 1 ] Ker(A - µl) when X # µ, and Y + E h ± , it follows that among the subspaces Y,; for different X there are only a finite number of degenerate ones. Let these be 2',;,, 2, ... , Y +K and let 9a = Yo,> + Y;,,,,, as

where'o,a, is the isotropic part of 9', , i.e., Yo,,,, = 2X fl .a;, and the subspace a, is definite and completes 2'o,x, into Ya . Since Lin (Yo,,,, } i is a finitedimensional subspace, it follows that dim .-W/f, (A) < co, where 61 (A) = C Lin (Y, , T,-, Y 1 ,,,, 21;>,, I µ ;4 Xi, i = 1, 2, ... , n ) is a non-degenerate subspace relative to A, and moreover the maximal (in f, (A)) uniformly definite subspaces 6'1± ( A ) = C Lin (2µ , Y;,,, I µ ;4 X1, i = 1, 2, ... , n) are invariant relative to A. We now note that the operators A I f i (A) satisfy the conditions of Lemma 2.7 relative to the scalar products ± [ , ] I f i (A) respectively,

and dim fl(A)I1] < oo, and moreover

fo(A I f,(A)I') and the

kernels Ker(A I 6'1(A)1'1 - XI) of the operators (A I e', (A) 1] - XI) are non-

degenerate. Consequently in each of the subspaces f; (A ), f,- (A ), and 91,, (A I f I (A ) [1) there are J-orthonormalized Riesz bases composed of eigen-

vectors of the operator A. The union of these bases will then be the required basis.

f) Before proving this assertion we point out that, if two bases differ on a

§2 Completeness and basicity of root vectors of J-dissipative operators 229 finite number of elements and if one of the bases is a p-basis, then the second one will also be a p-basis. In proving assertions d) and e) it was essentially established that if F (A) (respectively go(A )) coincides with 'W, then there is in .1Y an almost J-orthonormalized basis composed of root vectors (respectively,

eigenvectors) of the operator A, and all these vectors, with the exception,

perhaps, of a finite number, are characteristic vectors for A and lie in pre-assigned invariant subspaces Y+ E ff + n h ± of the operator A. Let this operator have the invariant subspace 2'+ E af+ fl h+ with the angular operator KY-, E .gyp, let (fc ) be the J-orthonormalized part of a Riesz basis

composed of root vectors (or eigenvectors) of the operator A, and let ( f, } C 2 + , where T_ = 2+ ('I. Since the system (f4 } U (fq } differs from a

basis in J on a finite number of elements, it can be constructed into a J-orthonormalized basis in ' which will differ from the original one on a finite number of elements. Therefore without loss of generality we shall suppose that ( and we shall prove that it is a U If. +,) is a J-orthonormalized basis in p-basis. Since (fa } U (ff } is a basis, we have 2 + = C Lin (ff }, and therefore 11 Ksr+ 11 < 1. From 2.Formula (5.3) we construct the operator U(Kv-,), which is positive, J-unitary, and such that U(Kv-+) - I E 19p. Since (f4 } U (ff } is a J-orthonormalized basis, (e.' } U ( e,-), where e« = U-1(Ky-.) fa , also is a J-orthonormalized basis. But since e, by construction lie in .jy + , it follows that (e,+) U (e.- } is an orthonormalized basis. It remains to notice that the inclusion U(Kv-,) - I E Yp implies the inclusion U-1(Kr.) - I E .9p, and therefore U { ff } is a p-basis.

Conversely, suppose there is in . an almost J-orthonormalized basis composed of root vectors (or eigenvectors) of A and that it is a p-basis. We shall prove that there is an invariant subspace Y+ E af+ fl h+ of the operator A with an angular operator K. E gyp. To do this we separate from the basis the J-orthonormalized system (ff } U if-,,) composed of the definite eigenvectors of the operator A, with (f+ } C j P± + . We form the subspaces C Lin (ff ). They are uniformly definite, invariant relative to A, and such that def Y') < co. By 3.Theorem 4.14 the dual pair (Y+", 9 1)) can be extended into a maximal dual pair (Y+, Y_) invariant relative to A. It is clear from the construction that _T+ E lf+ fl h+. We verify that E J'p. Since dim +/ 1) < co, we can suppose without loss of generality that .?'" = Y+, i.e., (f« } U is a J-othonormalized p-basis. Let f, = (I+ T)g±, where gq } is an orthonormalized basis in , T E Jp, and eq = U-' (Kv--) f, is an + . From 2.Formula orthonormalized basis in M composed of vectors ea E . (5.3) it can be seen that Ky-. E SPp if and only if U(K1-+) - T E 9'p. We

establish this lost result. Since e,± = U-'(Ky-.)(I+ T)ga , it follows that U-' (K,,.)(1 + T) = V is a unitary operator, and therefore I+ T= V(V*U(K,,-) V) is the polar decomposition of the operator I+ T. Hence, (I+ T*) (I+ T) = V*U2 (K,-_) V, and therefore U2 (K,--) - E .f p, which implies the inclusion (U(K,,.) + I)-' (U2 (K,-_) - 1) = U(K, -) - I E SP,-

0

230

4 Spectral Topics and Some Applications

Remark 2.13: Since completely continuous J-selfadjoint operators of the class K(H) and all a-selfadjoint operators with a spectrum having no more

than a countable set of points of condensation satisfy the conditions of Theorem 2.12, the assertions in it also hold for them. Moreover, for completely continuous operators it is possible, by Corollary 2.3, to write 1o(A) instead of Lin[Y),(A) I X E s(A)] in the formulation of Theorem 2.12. 4 In conclusion we investigate the question of the completeness of the system of root vectors of definitizable J-selfadjoint operators.

Lemma 2.14: Let A be a bounded J-non-negative operator having a spectrum with no more than a countable set of points of condensation. Then the equality '(A) =.Ye is equivalent to non-degeneracy of ?o(A).

Since 2'0(A) [1] 2,,(A) when µ ;e 0, it is clear that non-degeneracy of (A) (= i) implies non-degeneracy of Yo(A ). Conversely, let .?o (A) be non-degenerate. Then 6(A) is also nondegenerate. For, if ','(A) is degenerate and if xo E e (A) fl t (A)I1I, then [Axo, xo] = 0. Consequently xo E Ker A, which implies the degeneracy of

Yo(A).

Let us assume that 4"(A) ;d °, i.e.,

(0). We consider the

operator A' = A I e(A)WWW. This operator has the following properties: it is

positive relative to the G-metric [ , J I 6'(A)111; aa(A' I = 0; and a(A') consists of not more than a countable set of points. The first two of these assertions are trivial, and so we verify the third. To do this we note first that if X E p(A), then X E p(A'). From 2.Corollary 3.25 and Exercise 7 on §1 it follows that if X0(0) is an isolated point of the spectrum of the operator A, then the

range of values of the operator A - XOI is closed, Ker(A - XoI) is projectionally complete, and Ye = Ker(A - XoI) [+] Ker(A - XOI) W1l, and moreover Xo E p(A I Ker(A - XoI)W1). Hence we conclude that if Xo(#0) is an isolated

point in a(A), then XoEp(A'). By hypothesis v(A) has no more than a countable set of points of condensation and therefore a(A') consists of not more than a countable set of points. Let µo be an isolated point of the spectrum of the operator A', and let -y,, be

a circle of sufficiently small radius with centre at the point µo. Then dX is a G-orthogonal projector on to the 'G,,-space' Ae,,, = P,04" (A) [11 invariant relative to A', and a(A' I ,,0) = [µo) . The operator Aµ0 = A' W . is G,,,, positive, and µo 0 ap(A,,0). By 3.Lemma 3.16 we extend Aµ, into a f-non-negative operator Aµ, which has 14o as the only point of its spectrum. In accordance with Exercise 7 on § 1 Aµ0 = t ol,,0 when µo ;e 0 and A20 = 0 when µo = 0, i.e., µo E op(A')-but µp(A') = 0, so we have obtained a contradiction.

Let A be a bounded J-selfadjoint operator, defmitizable by the polynomial p(X) with the roots [X;) i and let o(A) have no more than a Theorem 2.15:

§2 Completeness and basicity of root vectors of J-dissipative operators 231

countable set of points of condensation. Then the equality ','(A) =,w is equivalent to non-degeneracy of Lin

Since f(A) = f (p(A)) and 9?o(p(A)) = Lin(2),,(A));, it is sufficient to use Lemma 2.14.

Remark 2.16: Since for a J-non-negative operator A the equality 9'),(A) = Ker(A - XI) holds for all X ;4 0, so, on replacing in the formulations of Lemma 2.14 and Theorem 2.15 the root subspaces by the corresponding

kernels Ker(A - XI) of the operator A, we obtain a criterion for the coincidence of fo(A) with W. In contrast to the case of operators of the class k(H) completeness of the system of root vectors or even of the eigenvectors of a J-non-negative operator

does not guarantee the existence of a basis composed of such vectors. However, the following theorem holds. Theorem 2.17:

Let a bounded J-non-negative operator A satisfy the condi-

tions of Lemma 2.14. Then Anx = EcX « [x, fa]fa sign X., where n > 2, (Xa) C ap(A), X. 3e- 0, and (fa) is a J-orthonormalized system composed of the eigenvectors of operator A: Afa = Xa fa; if moreover A is a J-positive operator, then Ax = Z.X. [x, fa] fa sign X. (here the series converge with respect to the norm of the space W). In accordance with Theorem 1.5 Ax = Sx + 17-P dE,x. From Lemma 2.7 we conclude that

E-x_-aO, [',fa)fa, a>a where (fa) is the J-orthonormalized system of eigenvectors of the operator A corresponding to the eigenvalues (Xa) : Af« = Xafa, Xa # 0. Consequently I - Ea=

17.P dE,x = EcXc [xifc] fa sign X,,, where the series converges with respect to the norm of ,' (we notice at once that if A is a J-positive operator, then S = 0 and therefore Ax = > c Xc [x, fa] f« sign X.\). We use Theorem 1.5:

Anx= An-`IS+ _

Xn. [x, fa ] f c Y

P

dE)x= An-1

sign X.

P dEx

when n = 2, 3, ....

Exercises and problems I

Let A be a completely continuous J-dissipative operator of the class K(H), let AjE %'j and limn-,, nsn (A) = 0. Prove that f(A) = .h" if and only if 'o(A) is non-degenerate (Azizov [4], [8], Azizov and Usvyatsova [2]).

4 Spectral Topics and Some Applications

232

Hint: Use Lemma 2.4 and the definite analogue of this assertion ([XI], Theorem 4.4.2). 2

when n - Co. Prove Let A E v'. fl K(H), OA = a1p, p > 1, and o(n that ('(A) = .h' if and only if'o(A) is non-degenerate [Azizov [4], [8], Azizov and Usvyatsova [2] ).

Hint: Use Exercise 6 on 2.§2, Lemma 2.4 and the definite analogue of this assertion ([XI], Theorem 5.6.1). 3

Let A E y'- fl K(H), OA = jr1p, p > 1, and for some a for the operator B = [e'°`A]r let s. (B) = o(n-'/ P) when n - oo. Prove that the equality f (A) =.1 is equivalent to the non-degeneracy of S'o(A) (Azizov [4], [8], Azizov and Usvyatsova [2]). Hint: The same as for Exercise 2 except that [XI] 5.6.1 is replaced by [XI] 5.6.2.

4

Let

5

Prove that if in a Pontryagin G`)-space Y there is at least one almost

be a Pontryagin G(')-space (i.e., 0 E p(G('))). Prove that there is in .W' at least one `almost G(''-orthonormalized p-basis' if and only if I G(') IE yP (Azizov and Kuznetsova [1]). G(')-orthonormalized p-basis, then any other almost G(')-orthonormalized basis is also a p-basis. In particular, if G(')2 = I, then any G(')-orthonormalized basis is a p-basis for any p > 0 (Azizov and Kuznetsova [1]).

6

Give an example of a J-non-negative bounded operator B ¢ 1. such that B2 E J'P (0 < p < oo). (I. Iokhvidov [8], Azizov and Shlyakman [1]). Hint: In 2 Example 3.36 put B, E .PP and take as B2 the linear homeomorphism mapping Y, on the .Y 2, B, > 0, B2 > 0-

7

Prove that if A is a bounded J-selfadjoint operator definitizable by the polynomial p(X) with the roots (k;);, then the following assertions are equivalent: a) C\(X1) 1 CP(A); b) there is an integer q > 0 such that AQp(A) E .f'm; c) Ap(A) E ,1 ,. (V. Shtraus [3]; Azizov and Shlyakman [1]).

8

Let A E .1 be a GI')-selfadjoint operator in W (0 5 x < oo), and let 0 E a,(A). Prove that then 411(A *) = .fit') and that there is in .W(') a Riesz basis relative to the norm I x II, =III GI 112X II composed of root vectors of the operator A ([31).

9

Let A E .y'm be a G-selfadjoint operator in W, let 0 E a, (A), and let at least one of the sets (- oo, 0) fl a(GA) or a(GA) fl (0, oo) consist of a finite number of normal

eigenvalues. Prove that then rr (A ") = W and that there is in # a Riesz basis relative to the norm II x II, =III GA I "2x II composed of root vectors of the operator A Q III] ).

Hint: Relative to the indefinite form [x, y], _ [Ax, y] the space .# is a G'-space with G(') = GA. Use the result of Exercise 8. 10

11

Let ,Y be a K")-space, with 0 E p (Wt' ), let J ' I Ker W(') be a Pontryagin space, and let A be a completely continuous W(')-selfadjoint operator. Prove that there is in .# a Riesz basis composed of root vectors of the operator A if and only if /'o(A) n C Lin(21a(A)I X ;d 0) = (B) (Azizov [7]).

Prove that if A E .y',0 is a r-selfadjoint operator and f (A) * IL, then it is impossible to choose in the subspace f (A) a Riesz basis composed of root vectors of the operator A (Azizov [7] ). Hint: Prove that the inequality OF (A) ;d n, implies the inequality V'o(A) n C Lin(.V',,(A) I X ;d 0) ;d [0), and use the result of Exercise 10.

§3 Examples and applications §3

233

Examples and applications

In this section we give some examples showing the impossibility of weakening the conditions in some of the theorems given in §2, and also 1

examples showing some applications of the results in the preceding section. For our first purpose we need the following. Theorem 3.1: Let A = (AR + iAj) be a continuous operator acting in a Pontryagin space II,,, and let AR and A, be a-non-negative operators. Then non-degeneracy of ?o(A) is equivalent to the inclusion Ker A n:3?A C wA.

Let Yo(A) be non-degenerate. In accordance with 2.Theorem 2.26 Yo (A) = A 'o [+] , ifo, where /{ "o is finite-dimensional and invariant relative to A, and -Ito C Ker A is non-degenerate. Consequently A"o is a non-degenerate subspace. By construction (see the proof of 2.Theorem 2.26) the kernel of the operator A, = A I .4'0 is neutral and it is the isotropic part of the kernel of the

operator A. Since A and A, are a-dissipative operators, it follows from 2.Corollary 2.17 that Ker A = Ker A`, and Ker A, = Ker A'1. This implies the equality Ker A fl 3A = Ker A fl (Ker A`) [11 = Ker A fl (Ker A) [11 = Ker A, = Ker A, n RA,.

Since RA, C RA, it follows that Ker A n 4A C A. Conversely, let Ker A fl.A C w A. If xo E Yo(A) n 2o(A )[1), then [Axo, xo] = 0. Using the fact that AR and AT are ir-non-negative we obtain that xo E Ker A. Consequently, since xo [1] £o(A ), we have xo [1] Ker A(= Ker A`), and therefore xo EJIA, i.e., xo E Ker A n 4A. By hypothesis Ker A n 4A C RA, and therefore there is a vector yo such that xo = Ayo. The vector yo E '0(A) and therefore 0 = [xo, yo] = [Ayo, yo]

Again using the fact that AR and AI are 7r-non-negative, we obtain that (Ayo = )xo = 0, i.e., 91o(A) is non-degenerate.

The theorem just proved enables us easily to construct examples demonstrating that in the theorems about completeness (see §2) the condition of non-degeneracy of Lin [Yx(A) I X E s(A )] does not follow from any of the other conditions. We give one such example, and leave the reader to construct others. Example 3.2: (cf. Corollary 2.9 and Exercise 3 on §2 when p > 2). Let B be a nuclear operator in an infinite-dimensional space W, with Ker B ;4 (0), and let

the operators i (B+ B*) and (I/2i)(B- B*) be non-negative. In Ker B and B I RB we fix on vectors xo (11 xo

= 1) and yo (11 Yo I I = 1) respectively. Since

B is a dissipative operator, xo 1 yo. This in turn implies that the operator

J: J(«xo+ayo)=axo+ayo,

JI(Lin(xo,YoI)` = -I

234

4 Spectral Topics and Some Applications

is selfadjoint and unitary. By means of this operator we introduce the form [x, y] = (Jx, y), turning the space ,Y into a Pontryagin space with one positive

square. The operator A = JB is nuclear, and AR = J[(B+ B*)/2] and Aj = J[(B - B*)/2i], and so AR and A, are ir-non-negative operators. Since Ker A = Ker B and yo 1 Ker B, so xo [ 1 ] Ker A, i.e., xo is an isotropic vector in Ker A, and therefore xo E Ker A n A. But xo 0 ,3 A, for otherwise yo would be in ?B. Consequently, by Theorem 3.1, Yo(A) is degenerate. 2 The following example shows that in Theorem 2.8 the condition that the operator A be 7r-dissipative cannot be replaced by the more general condition that A is a J-dissipative operator of the class H.

Example 3.3: A Volterra J-dissipative operator A of the class H with a nuclear J-imaginary component and sp AI = 0. Let W = W' O+ .,Y- be a J-space, where W ± L2(0, 1). We put A;i = 0 when i P6 j,

A = II AjiMI ?i= 1,

All = -A22=2i I

r

ds

As is well-known (see, e.g., [XI], 4.7.4) A11 is a Volterra disipative operator

and (112i)(A11 - Ail) is a one-dimensional operator. So A is a Volterra J-dissipative operator with a two-dimensional J-imaginary component AI and sp AI = sp(A 11)i + sp(A22 )J = 0. It remains to verify that A E H. To do this it

± are its only maximal semi-definite invariant suffices to establish that subspaces. In the present case the latter is equivalent to the fact that for any it follows from BA 11 = A22B that B=O (see 3. Lemma 2.2). Let 2iBJ0 ' f(s) ds= -2iSo (Bf)(s) ds. Differentiating both sides of this

bounded operator B

equality with respect to t we obtain Bf = - Bf, i.e., Bf = 0 and therefore B = 0.

3 We recall that a function K(s, t) of two variables defined on a square [a, b] x [a, b] is called a Hermitian non-negative kernel if for any finite set of points (t1) i of [a, b] and for any complex numbers (;) i the sum E"J_,K(t;, t;);; is non negative, and it is called a Hermitian positive kernel if Ej",j= jK(tj, t;)E; ; = 0 if and only if , = 0, i = 1, 2, ... , n (for a more general definition see 4.§3.11 below). The function K(s, t) is called a dissipative kernel if (1/2i)(K(s, t) - K(t, s)) is a Hermitian non-negative kernel, and a strictly dissipative kernel if (1/2i)(K(s, t) - K(t, s)) is a Hermitian positive kernel. Let 9 = L2(a, b) (cf. Exercise 5 on 1.§2), i.e., the space of all w-measurable

functions sp such that lab yp(t)I2 dw(t)I < oo, where w is a function either non-decreasing or non-increasing on [a, b], and the scalar product (,p, >G) is given, up to sign, by the relation (,p, >G) = Jo p(t)>G(t) dw(t). Moreover, we I

§3 Examples and applications regard p =

235

if

Jb I P(t)-0(t)12 dw(t) = 0 a

Let a(t) be a function of bounded variation on [a, b]. In Exercises 5 and 6 on 1.§2 it was proved that the space L2, (a, b), where w(t) = WI (t) + W2 (t), and

a(t) = wi (t) - u)2 (t) is the canonical representation of a(t) in the form of the difference of two non-decreasing functions, and the space is provided with the do(t) is a J-space. In particular, if wi(t) is a piecewiseconstant function with x points of growth, then b) is a Pontryagin space with x positive squares (see Exercise 7 on 3.§9). In this and the following paragraphs we apply the results obtained in §2 to the investigation of integral operators A = Jb, K(s, t) do(t). form [gyp, '] = JQ

3.4: Let K(s, t) be a dissipative kernel continuous on [a, b] x [a, b], let A and A, be integral operators defined by the relations

Theorem

A = L K(s, t) da(t),

A, =

K(s, t) dt, J aa

a

let o(t) = WI M - w2(t), and let w, (t) be a piecewise-constant function with x

points of growth. Then if the system of root vectors of the operator A, corresponding to its non-zero eigenvalues is complete in C[a, b], then the system of root vectors of the operator A corresponding to its non-zero eigenvalues is complete in b). If in addition a(t) has no intervals of constancy, then the system mentioned of root vectors of the operator A is complete in C [a, b].

It follows from the definition of the operator A that it is a nuclear 7r-dissipative operator. Therefore by virtue of Corollary 2.9 the root vectors of the operator A corresponding to its non-zero eigenvalues will be complete in L2,(a, b) if and only if 0 ¢ ap(A). We prove that 0 0 ap(A). Let cpo E Ker A. Then po E Ker A`, i.e., JaK(s, t)(po(s) do(s) = 0. Let 1(t), ("2(t), ... be the root vectors of the operator A, corresponding to the non-zero eigenvalues and forming a system complete in C[a, b]. It follows from the definition of root

vectors that ;(t) E MA,, i.e., there are functions ti(t) E C[a, b] such that (t) = (AIE;)(t) (i = 1, 2, ...). Then b

[0o, ]'i] =

b

po(s)(AjEj)(s) du(s)

soo(s)(";(s) du(s) = J a

J a

b

a

b

'PO (s) J

K(s, t)E;(t) dt da(s) a

b

b

;(t) a

K(s, t),po(s) du(s) dt = 0. a

Since C[a, b] is denose in

b), we have po = B, i.e., 0 ¢ ap(A).

4 Spectral Topics and Some Applications

236

Now let a(t) have no intervals of constancy, and let -q,,'92, ... be a system, complete in y (a, b), of root vectors of the operator A. Since q j E A, the ?7i are continuous functions, and in b) there are functions > i such that n; = Ai,&; (i = 1, 2, ...). Let 4' be a continuous linear functional on C[a, b] such

that 4'(,q;) = 0 (i = 1, 2, ...). By virtue of Riez's theorem on the integral representation of a linear functional on C[a, b] there is a complex function of bounded variation CD such that (AOi)(s) dw(s)

4)(AO;) = J b a

= r b rb

K(s, t)O;(t) da(t) d(5(s) a

a

= J ba Oi(t)

a a

(i = 1, 2, ...)

K(s, t) &Z(s) du(t)

b), and the function w = wl + wz like a has no intervals JoK(s, t) d(,)(s) = 0 for all t E [a, b]. But then Ja ;(t) Ja K(s, t) d(:w(s) dt = 0 (i = 1, 2, ...). Since [ j';} is complete

Since if (A) = of constancy,

in C[a, b] we conclude that 4i = 0. This is equivalent to the completeness of (,t;} in C[a, b]. 4

Now let K(s, t) be a Hermitian positive kernel, and let a(t) be an

arbitrary function of bounded variation on [a, b]. We bring into consideration the iterated kernels K(")(s, t) = K(s, t), K(n) (s, t) = fab K(n-,) (s, l)K(l, t) da(l) (n = 2, 3, ... J

Moreover, we assume that the kernel K(s, t) generates a bounded operator A = JQK(s, t) da(t) having no more than a countable set of points of condensation of the spectrum. For example, if I K(s, t) 15 c < oo and if the function K(s, t) is continuous in each variable when the other is fixed, then (see, e.g., I. Iokhvidov and Ektov [1], [2]) A is a completely continuous operator and therefore has a single point of condensation of the spectrum. Let 71, denote the eigenvectors of the operator A corresponding to X. Re 0. In accordance with Theorem 2.17 for every function xE L2. (a, b) we have

A"x=

Xn [x,,7.],a(t)sign X. LT

b

_ E va' ?a (t) a

X(T)17,(T) da(T)sign X.. a

The function K(")(s, t) belongs to

b) with respect to each of the

variables. Consequently b

X_

AK(s, t) a

a

X.,)a(t),ta(s)sign X.. a

§3 Examples and applications Similarly AKA") (s, t)

X.

237

(n = 2, 3, ...

a

Here the series converge in the norm of L2,(a, b). We note also that, by definition, K"") = AK("-'). So we have proved

Theorem 3.5: If a Hermitian positive kernel K(s, t) and its iterations K(") (s, t) belong to b), and if the operator A = J K(s, t) da(t) is continuous with not more than a countable set of points of condensation of the spectrum, then when n > 2 Kt"t (s, t) _

X?7a(t),la(s)sign Xa,

(3.1)

a

where (na) is a J-orthonormalized system of eigenvectors of the operator A corresponding to Xa # 0, and the series (3.1) converges in the norm of the

space L2a, b). Theorem 3.5 has been obtained as a simple consequence of Theorem 2.17. By applying additional and different methods, going beyond the scope of our

book, for an integral operator and iterations of the kernels generated by it more precise results can be obtained (see, for example, Exercises 3-5 below).

5

In S. Krein's article [1] it is shown that the problem of the oscillations of a

heavy viscous fluid in an open fixed container reduces to the study of an operator-valued function

L(X)=XG+C-'H-I,

(3.2)

where G and H are completely continuous selfadjoint operators of finite order, i.e. G E 91P, HE 9q, p, q < oo, and G > 0, and H > 0. A whole series of general non-selfadjoint boundary problems with a parameter X in the equation and in the boundary conditions reduce to the spectral analysis of a similar operator-valued function. Here we show one of the ways of analyzing an equation (3.2) based on applying Theorem 2.13. For other results in this direction see Exercises 6 and 7 below.

Definition 3.6: In equation (3.2) let G = G ` and H = H* be bounded operates acting in a Hilbert space .'. A point Xo E C is said to be a regular

point for the function L(X) if 0Ep(L(Xo)); otherwise it is a point of the spectrum of the function L(a). A vector xo is called on eigenvector of the function L(X) if there is a Xo E C (Xo is an eigenvalue) such that L(Xo)xo = 0. The vectors x,, x2.... , x," are said to be associated with the eigenvectorxo and

the set (x;)o' is called a Jordan chain if CYJL(X0)

Z j!ax j xk-l = B ,=o

(k = 0, 1, 2, ... , m).

4 Spectral Topics and Some Applications

238

Following M. Krein and Langer [2] we introduce a scheme of argumentation which reduces the spectral analysis of the function L(X) to the spectral

analysis of a certain J-selfadjoint operator. We shall suppose that G > 0. After replacement of the variables X _ - µ ' - a the function L (X) becomes the function

L,(µ)

µ(1 +µa)

[µ2(a2G+ H+ aI)+µ(2aG+ I) + G].

We put

a>inf(b>01 Fb= b2G+H+b1>0). Then

Li(µ)= - µ(1 +µa) Fa (µ2l+µBa+Ca)FQ/2, 1

where

Ba=Fa "2(2aG+I)FQ v2)> 0,

=Fa'2GFa''2

Ca

It can be verified immediately that Xo is an eigenvalue of the function L(X) if and only if µo = - (Xo + a) - ' is an eigenvalue of the function L2(µ) =,U21+ µBa + Ca, called a quadratic bundle. Moreover, (xo, xi, ... , is a Jordan chain of the function L(X) if and only if is a Jordan chain of the bundle L2(µ). (FV2xo, FcV2x,, ... ,

_ .i+ O+ e-,

We bring into consideration the J-space the J-selfadjoint operator 0 Caln

Aa =

Ve

_' and

CQi2l (3.3)

-Ba

acting in it. It is easy to see that the regular points, the spectrum, and in

particular the eigenvalues of the bundle L2(µ) and of the operator Aa coincide. Moreover, if (yo, yl, . . . , y.,.) is the Jordan chain of the bundle L2(µ) corresponding to the eigenvalue µo, then the vectors Co /2 Yo

C.1/2 Y1

ILOyo

µ0y I + Yo

Cav2

µoyn, + yin- I

form a Jordan chain of the operator Aa, and conversely if zi') zf2)

is a Jordan chain of the operator A., then yo=

I AO

z62),

(z(?)yi= 1 (zf2)- yo),..., y,,,= 1 µo µo

is a Jordan chain of the bundle L2(µ).

y,,,-I)

§3 Examples and applications

239

Thus, a one-to-one correspondence has been established between the Jordan

chains of the function L(X) and the operator A. Here we denote by symbols .4->, and -lix

KerL(X) fl ((2XG- I)Ker L(X))1

KerL(X)fl (Lin((2XG-I)xk+Gxk_,)o")1 respectively, where (xo, x,, ... , x,,,) are all possible Jordan chains of the and

bundle L (X) corresponding to the eigenvalue X, and x_ 1 = 0.

Let the function (3.2) be given, where G > 0, G E 99.,, Theorem 3.7: H = H* is a bounded operator, and the set a(H) is no more than countable. Then:

1) each of the operators (3.3) generated by the function L (X) belongs to the class H;

2) dim .'/f(Ao) < dim Y16-o(A,,) < oo; 3) fo(AQ) =.W'

if and

only

(211G11)-'IxIs211H11;

4) f (A0) = 1Y if and (211 GII)-'

only I XI s211H11;

if

.

"\ = (0)

for all

X

such

that

if

/1"a = (0)

for all

X

such

that

5) If fo(A.) = W (respectively, f (Aa) = M), then there is in W an almost J-orthonormalized Riesz basis composed of eigenvectors (respectively, root vectors) of the operator Aa. If fo(A,,) = X, then there is in W a J-orthonormalized Riesz basis composed of eigenvectors of the operator

A. if and only if the function L(X) has no non-real eigenvalues. Moreover, the bases mentioned above can be chosen as p-bases if and only if G E 9'p12-

1) Since it follows from G E Y. that Co/2 E 9'- so (cf. 3.Theorem 1.13, El 2.Remark 6.14, and 3.Theorem 2.8) the operator A has at least one maximal

non-negative invariant subspace 9+ with an angular operator K. In accordance with 3.Lemma 2.2 Kr--CQ"2K, + C"2 + B,,K'-- = 0, and since B,, )> 0, so Kv~ = - BQ'(CQ"2 + K,-CQ,2K,--) E q.. Consequently (cf. Exercise 3 on 3.§5), A E H, and moreover, Kv,. E .%'p if and only if CQ12 E 9p, which

in turn is equivalent to G belonging to the class .9'p,2 (see [XI], 3.7.3). Assertions 2) to 5) will now follow from Theorem 2.13 if we show that

the set a(Ao) is no more than countable, that the non-real eigenvalues of

the

operator

Aa

and

the

set

(t=-(X+a)-' 1(211G1-'slxl<211H11),

s(AQ)

lie

the set ./VX=(0) and in

and that = (0) are equivalent to non-degeneracy of Ker(A,, - µI) + Ker(A,, - AI) and 2 (AQ) respectively when µ = (X + a)-'. That the set v(A,,) is no more than countable follows from the fact that the operator Ao is the sum of the operator 0

0

0

- Fo '

240

4 Spectral Topics and Some Applications

and the completely continuous operator Ca z

0

- 2 aFa 1/2 GFa

- cJ"2

1"2

and the spectrum of the first of these consists of not more than a countable

set of points, since the operator H, and therefore also the operator Fa = a 2 G + aI + H, has this property (see 2.Theorem 2.11). We now verify that all the non-real points of the spectrum of the function Y'(X) are situated in the ring (2 II G II)-' S I X I S 2 II H II . For, if L (Xo)xo = 0 and Xo ;x-I o, then, solving the quadratic equation X o(Gxo, xo) - Xo(xo, xo) + (Hxo, xo) = 0,

we obtain Xo =

(xo, xo) ± ,(xo, xo)T- 4(Gxo, xo)H(xo, xo) 2(Gxo, xo)

Hence,

I x0 2- (Hxo,xo)-2(Hxo,xa). (xo,xo) (Gxo, xo)

2(Gxo, xo)

(xo, xo)

But since the discriminant of the equation is less than zero, we have (xo, xo) < 2(Hxo, xo) 2(Gxaxo)

(xo, xo)

and therefore

(xo, xo) 1 z\ I Xo I z\ 2(Hxo, xoz `2(Gxo, xo)J

(xo, xo)

Hence, we conclude that X0 is situated in the ring (2 II G II) -' S I X I S 2 II H II Consequently the non-real spectrum of the operator A coincides with points µo

of the form µo = - (Xo + a)-'.

We now show that the set a(Aa) is situated in the same ring. Let µ, = µ, E up(Aa) and let Ker(Aa - µI) be degenerate. As proved earlier this is µi ' - a) a vector equivalent to the fact that there exists in Ker L (X,) (X, xo such that the vector CaizF./Zxo

-

1

+a

F/2xo

is J-orthogonal to all vectors of the form Caiz1vzy

,+a Fvzy 1

§3 Examples and applications

241

where y E Ker L(X1), i.e., (Ca/ 2Fa"2xo, Ca/2Fo'/2y)

- (X1 + a) 1

2

(Fa12xo, Fa12y)=0.

Starting from the definitions of the operators C. and F. we obtain as an equivalent form of this equation

((X G + 2aX, G - I- H)xo, y) = 0.

(3.4)

We substitute in this equation the vector Xtxo - X1Gxo instead of the vector

Hxo and, after cancellation by X1 + a we obtain ((2X 1 G - I)xo, y = 0. In particular, when y = xo, we have X = (xo, xo)/2(Gxo, xo). But if in (3.4) we substitute instead of X i Gxo the vector X 1 xo - Hxo and instead of X 1 Gxo the

vector xo - (l/X)Hxo, and cancel out X, + a (when y = xo) we obtain that X, = 2(Hxo, xo)/(xo, xo). From this we conclude as before that X, lies in the

ring (211 GII) 'S IXI <2IIHI1. The validity of the interpretation given above of the equalities,'Ux = (B) and ..ffa = (0) is proved by entirely similar considerations.

Remark 3.8: In the proof of the equality f (A.) = , it is necessary to verify the triviality of Ji, only for those points of s(AQ) which are points of condensation of the spectrum of the operator Aa. But the latter coincide with the set of points of the form µ = - (X + a) - ' where X traverses the set of points of condensation of the spectrum of the function (3.2). If G, HE .9pe, then it

follows from the form of the operator A. that

it

has two points of

condensation of its spectrum : µ, = 0 and µ2 = - a - '. Consequently the function (3.2) also has only two points of condensation of its spectrum: X1 = 00 and X2 = 0 respectively. Both these points do not lie in the ring (211 G I I ) - ' S I X 15 2 II HII, and therefore when G, HE 9'm we always have

f(A)='.

Definition 3.9: The function L (X) = X G + (1 / X )H - I is called a strongly damped bundle if (x, x)2 > 4(Gx, x)(Hx, x) for all x ;e 0. Corollary 3.10: If under the conditions of Theorem 3.7 the function (3.2) is a strongly damped bundle, then there is in W a J-orthonormalized Riesz basis

composed of eigenvectors of the operator A, and it will be a p-basis if GE 1'p,2.

By Theorem 3.7 it suffices to verify that the function (3.2) has no non-real eigenvectors and that A'A = (0) for all X. The first follows immediately from

the fact that if X is an eigenvalue of the function (3.2) and if xo is the corresponding eigenvector, then Xo =

(xo, xo) ± .(xo, xo) - 4(Gxo, xo)(Hxo, xo) _ -o 2(Gxo, xo)

4 Spectral Topics and Some Applications

242

Let us suppose that for some Xo the subspace ,/I'>, ;e (0). Then, in particular, some xo E Ker ((2Xo G - I)xo, xo) = 0 for L (Xo), and therefore Xo = (xo, xo)/2(Gxo, xo), which in turn implies the equality (xo, xo)z =

4(Gxo, xo)(Hxo, xo), and we have a contradiction with the fact that the function (3.2) is strongly damped.

Exercises and problems 1

2

Investigate the possibility of generalizing Theorem 3.1 to the case of a Krein space.

let K(s, t) be a Hermitian positive kernel continuous on the square 0 5 s < t < b, let R(s, t) be a function bounded on this same square and continuous with respect to each variable when the other is fixed, let R(s, t) = R(s, t), and let a(t) generate a

Pontryagin space according to the form [,o, ,&J = la do(t), and let A = Jo (R(s, t) + iK(s, t)) do(t). Prove that the equalities F (A) = L2 (a, b) and JQ K(t, t) do(t) = E; Im k;, where (a;) is the set of eigenvalues of the operator A, are equivalent. (Azizov [8] ). 3

Consider on [a, b] x [a, b] a Hermitian bounded kernel K(s, t) continuous with respect to each variable when the other is fixed, and a function of bounded variation

a(t) 0 const. We shall say that the kernel K(s, 1) is a-non-negative if K(s, t)g(s)g(t) da(s) da(t) >, 0 for all g E a

Y,

a

where )/" is the lineal of all those functions from L2 (a, b) (o = Var a) which are generated by continuous functions from C[a, b]. Prove that a-non-negativity is equivalent to either of the conditions: a) J b K(s, t) dr(s) dr(t) > 0 for any (complex) function of bounded variation r on [a, b] with E, C Ea (E, and E. are the sets of points of variation of the functions r and a respectively); b) Y- k=, K(t;, tu)Eksk ,>O for all n E N, ti, t2, .. ,,, E Ea, and

(I. Iokhvidov and Ektov [1], [2]). 4

Let K(s, t) be a a-non-negative kernel satisfying the conditions of Exercise 3. Prove that the assertions `K(2 (s, t) = 0 on E. > Ea' and 'K(2 (s, t) = 0 on [a, b] x [a, b]' are equivalent (I. Iokhvidov and Ektov [1], [2]).

5

Prove that the integral equation p(s) = XJo K(s, t),p(t) da(t) with a a function of bounded variation, and a a-positive kernel, has x < oo positive characteristic points if and only if the positive variation of the function a has exactly x points of growth (M. Krein [1], I. Iokvidov and Ektov [1]).

6

Let S=S*E.y'., T=T`E.y'm, -1Ep(s),0>iao(T)andA=(I+S)T.Prove that 6 (A) = .0 and that in i' there is a Riesz basis composed of root vectors of the operator A ([XI] ); if S E .vo, then there is in .W' a p-basis composed of root vectors

of the operator A (Kopachevskiy [1]-[3]). Hint: The operator A is 7r-selfadjoint relative to the form [ It only remains to use Remark 2.13 and Exercise 4 on §2. 7

,

] _ ((I+ S)

'

,

).

Prove that if L (k) is the function (3.2), G = G * E .v'm, H = H* E .'/'. and Ker G =

Ker H= (0), then:

Remarks and bibliographical indications on Chapter 4

243

1) the system Xk(k)

{

(-ly k

j=0

j+1 xk-j(t)

1 of `special vectors', where (xo(X), ..., xk(k) ( is the Jordan chain of the function L(X) corresponding to the eigenvalues k, is complete in .JY (Askerov, S. Krein and Laptev [ 1 ] );

2) in ,Y there is a Riesz basis composed of vectors of the special form (Larionov [6], [7]); 3) If G E yo, HE 'Q, and r = max(p, q), then there is in -Y an r-basis composed vectors of the special form (Kopachevskiy [1]-[3]).

Remarks and bibliographical indications on chapter IV §1.1. The J-spectral function was introduced by M. Krein and Langer [1] for the case of IL, and later Langer [4], [5] transferred the investigation to the case of Krein spaces. The definitions in the text, given by Azizov, modify Krein and Langer's definitions in order to accommodate them to operators of the K(H) class (see Exercises 1 and 2). §1.2-1.3. Definitizable operators in H. were introduced by I. lokhvidov and

M. Krein [XV], and in a J-space by Langer [4], [5], [9]. All the results of these sections are due to him. We borrowed from Bognar [5] the elegant proof of Theorem 1.5. In Bognar [5] these are also given bibliographical references

to other proofs and, in particular, references to M. Krein and Shmul'yan's proof [3] based on considerations from the problem of moments. In connection with criteria for the regularity of the J-spectral function see Langer [4], [5), Jonas [1]-[7], Jonas and Langer [1], [2], Akopyan [1]-[3], and Spitovskiy [ 1].

§ 1.4. Definition 1.11 and Theorem 1.12 for the case of operators acting from one space into another are modifications of corresponding results of Potapov [ 1], Ginzburg [2], M. Krein and Shmul'yan [3]. On square roots of J-selfadjoint operators see Bognar [ 1] and, more fully, Bayasgalan [ 1]. §2.1. The results in this paragraph are due to Azizov. Corollary 2.3 was published in the paper by Azizov and Usvyatsova [2]. §2.2. Lemmas 2.4 and 2.5, Theorem 2.6 and Corollary 2.9 in the case A E Y. were published in the paper by Azizov and Usvyatsova [2]. The presentation in the text is due to Azizov. To him also is due Theorem 2.8, which is a transfer to IIX of a corresponding result from [XI]. Lemma 2.7 is, apparently, well-known, though we have not found its formulation in print.

§2.3. Theorem 2.12 is due to Azizov [13]. For the formulation of the problems about p-basicity he is obliged to Kopachevskiy, who first began the

study of this question for the case of indefinite spaces in connection with problems of hydrodynamics (see, e.g., Kopachevskry [ 1]- [3]). The concept of p-basicity itself was introduced by Prigorskiy [ 1]. We mention also that the

244

4 Spectral Topics and Some Applications

questions of completeness of a system of root vectors of 7r-selfadjoint operators A E .`/'- with Of aa(A) were investigated for the first time by 1. lokhvidov [2]; the existence of a Riesz basis of root vectors of such operators was proved essentially in [XI]; the criterion for completeness and basicity of these vectors without the condition 0 0 ap(A) was given by Azizov and I. Iokhvidov [1]. On other conditions for completeness and basicity and for a historical survey see [IV] and also the Exercises. §2.4. The results of this paragraph in so general a formulation are due to Azizov. In the case when the set a(A) has no more than a finite number of points of condensation, Theorem 2.15 and Remark 2.16 (even in the case of Banach spaces with an indefinite metric) were obtained earlier by Azizov and Shtraus [ 1 ] , and Theorem 2.17 when A 2 E 9 " . is due to Kuhne [ 1 ] (see 1. lokhvidov [ 8], Ektov [ 1 ] ). As above, for a historical survey and for other results we refer the reader to the survey [IV] and to the Exercises on this section.

§3.1-3.3. All the results in these paragraphs are due to Azizov. Some of them were published in Azizov's paper [8]. Theorem 3.4 for the case of a Hermitian kernel was obtained earlier by I. Iokhvidov [2]. §3.4. Theorem 3.5 in the text was proved by Azizov; for the case A E 3'm see M. Krein [1], I. Iokhvidov and Ektov [1]. §3.5. Theorem 3.7 and Corollary 3.10 were proved by Azizov [13] (cf.

Askerov, S. Krein and Laptev [1], M. Krein and Langer [2], Larionov [6], [7], Kopachevskry [1]-[3], Azizov and Usvyatsova [2]). As regards other investigations of operator bundles and application, of an indefinite metric see Langer [4], [6], [10], [12], Kostyuchenko and Orazov [ 1 ] , and others.

5 THEORY OF EXTENSIONS OF ISOMETRIC AND SYMMETRIC OPERATORS IN SPACES WITH AN INDEFINITE METRIC

In § 1 the apparatus of Potapov-Ginzburg transformations is developed, and its application to the theory of extensions is demonstrated. §2 is devoted to another approach to the theory of extensions of isometric operators in Krein spaces. In §3 generalized resolvents of J-symmetric operators are described.

§1

Potapov-Ginzburg linear-fractional transformations and extensions of operators

One of the methods allowing extensions of (Jr, J2)-isometric operators to be constructed is the application of Potapov-Ginzburg transformations or, 1

briefly, PG-transformations. Definition 1.1: Let .,Y, =X1' Q+ .YP,- and .YP2 = .02' Q+

. z be the canonical decompositions of a J,-space ., and a J2-space .02 respectively, and let 1(3 = .Wi O+ )Yz and ,Y4 = ,Yz OO i"i be a J3-space and a J4-space constructed from them with J3 = It Q+ - Iz and J4 = Iz (j - IF. A transformation w+ :.r°1 O+ .#2 -- H3 .W'4 carrying a vector (x,, x2) into a vector

245

5 Theory of Extensions of Isometric and Symmetric Operators

246

(X3, X4), where

XI=Xj +Xl

X2=X2 + X 2

,

X3=X1 +X2

,

,

X4=X2 +X1

,

,+ (i = 1, 2), is called a PG-transformation. E As well as the transformation w+ having the form

x,

w+:

1 QIY2--,IY3 Q+'4,

W

other PG-transformations can be considered: U)

:.Y1 G. 02-x,03 Q W4,

W±:

w (XI,X2)= <-X1 +X2,X2 +X1 ), W±<X1,X2)=(X2 ±X1,±XI +X2).

-W2-J4('Y3

Below we shall study the properties of the transformation w+. The properties of w- and w+ are similar, and we leave it to the reader to formulate them and prove them by the same scheme as for w+ or relying on the following obvious proposition: 1.2.

W- = w+J,

w- (X1, X2) = (w± (x1, x2))-1,

CO- = W+J,

where

J(x,y) = (-x, y)

and

((x,Y))-' = (Y, X).

We note also that the spaces W3, ,4 and i, M2 may interchange roles, and the transformation w+ can be applied to a vector <x3i x4) E. 3 ( ,W'4. In

what follows we shall cite definitions and propositions for one of these variants, remembering that they are symmetric.

1.3. The transformation w+ maps ,Y, O+ .2 one-to-one on to .3 (D and it is involutory: w+(w+<xl, x2)) = <xl, x2), and if (X1, X2),
4,

and (X3, X4) = w+<X1, X2), (Y3, Y4) = W+(Y1, Y2

then

[xl, Y1 I I - [X2, Y212 = (X3, Y3)3 - (X4, Y4)4

here ( , )3 and (

,

)4 denote the scalar products in .7(3 and

4 respec-

tively.

The vector < X3, X4) = w+ < X1, X2) is defined uniquely by the pair < xi, x2)

because projection of xi on to Wi± (i = 1, 2) is single-valued. The existence of the inverse transformation, its definition on the whole space .'Y3 O+ .7(4, and the equality w+(W+<x1, x2) are obvious. It remains to verify the identity (1.1): [x1,Y1]1- [x2,Y2]2=(x1+,Y1 )I-(X17, Y? )1-(X2+, Y2 )2+(x2 ,Yz )2

=(Xi +xz ,Y1 +yz )3-(x2+ +x1-,Y2 +Yi )4 = (x3, Y3)3 - (X4, Y4)4.

§1 Potapov-Ginzburg linear fractional transformations

247

2 We define the PG-transformation w+ on sets ./VC .1 O+ .02: (1.2)

w+(.4) = (w+<xl, x2) I <x1, x2) E.4].

From Definition 1.1 and formula (1.2) it follows immediately that 1.4 If 2' is a lineal in '1 O+ .2, then w+ (Y) is a lineal in A3 + .r4. El

Definition 1.5: Let T :.'1 -+ .02 be a linear operator. We shall say that T E T if Ker(PZ TPi I.3 'j ) = (B) , and we introduce the PG-transformation

w+ (T) :3 -'4 for such operators by the formula w+ (T) _ (Pi + PZ T)(Pi + PZ T)-

(1.3)

(the existence of the operator (Pi + Pz T)-' follows from Exercise 6 on 2.§1). Theorem 1.6:

If the operator TE J and if rr is its graph, then

w+(rr)=r,-(r) and w+(T)E T. Since 9,,-(T)= ((Pi + Pz T)x1 I xl E fir) and FT= (<xl, Tx1) I xl E cT), we have, using (1.2) and the Definition 1.1, w+(rr) = (w+<x1, Tx1) I x1 E VTJ

_ (((P; + Pz T )xl, (Pi + PZ T )xl) I X1 E CAT) _ (<(Y3, (Pi + P3 T)(Pi + Pj T)-'Y3) I Y3 = (Pi + Pz )xl, x1 E !2T) = r.+ (r).

Since w+ (T) acts from .)r3 into W4, so, in accordance with Definition 1.5 and Exercise 6 on 2.§1, w+ (T) E J if Ker(P3 + Pa w+(T)) = (0). Let

Y3 E Ker(P3 + Pa w+ (T));

Y3 = (Pi + Pj T)x1 E cA-(r), x1 E CAr.

Then

0=(P3 +P4w+(T))y3=P3(Pi +Pi T)x1+Pa(PI +P3T)x1 = xi + ...XI = x1. Consequently,

y3 = 0,

i.e.,

Ker(P3 + Pa w+ (T)) = (0),

and

therefore

w(T)E T. Corollary 1.7: The transformation (1.3) establishes a one-to-one correspondence between (J1, J2)-non-expansive operators V:.11 -.y'2 and con-

tractions W:.3 -p .4, WE T. Moreover, V is a (J1, J2)-bi-non-expansive operator (respectively, a (J, J2)-isometric operator; a (J1, J2)-semi-unitary

operator; a (J1,J2)-unitary operator) if and only if W=w+ (V) is a contraction, V w =

.

3 and R p,- wP_ = .01 (respectively, W is an isometry; W is

an isometry and gyp,- wpb =i ; W is a unitary operator and 5?p,- wp, =,01).

248

5 Theory of Extensions of Isometric and Symmetric Operators

It follows from Exercise 33 on 2.§4 in conjunction with 2. Remark 4.29 that strict minus-operators (including (J1, J2 )-non-expansive operators) VE J. It only remains to use Theorem 1.6, the Formulae (1.1) and (1.3), the

definitions of the corresponding classes of operators, and the results of Exercise 33 on 2.§4 and Exercise 7 on 2.§1. For, let us carry out, for example, the appropriate argument for the case when V is a (J1, J2)-unitary operator. It follows from Theorem 1.6 that WE J. From (1.1) we conclude that W is an isometry, and from (1.3) in combination with Exercise 33 on 2.§4 that 9 w = IY3. Similar considerations show that the domain of definition of the operator

w+(V)

(P+ +PZ V) (Pi +Pz V)-':4-.3

.4. It remains to observe that (cf. Proposition 1.2) w+ (V) _ (w+ (V)) ' = W-', and therefore W is a unitary operator. Since Pi WPz I . z = (Pi VPi IM (see Exercise 1 below), ?p,- wP_ = . i Conversely, let W be a unitary operator and let 91P,- wp, = W_-. One proves, is

as we did above for W, that the operator V is defined on '1, and V is a (J1, J2)-isometric operator, i.e., V is a (J1, J2)-semi-unitary operator. From the facts that the operator WE J is unitary and that 91P,- wP. = ,Y1 it is obtained immediately that

Ker(P2 WPi I.W )2 = [e1,

J?P, wp, = w2+,

and, using the same considerations as above for W, we obtain that the operator V -' is (J1, J2)-semi-unitary, and therefore V is a (J1, J2)-unitary operator. IY2. We recall that an 3 Again let rT be the graph of the operator T:.Yf 1 operator Tis an extension of the operator Tif and only if rT C Pr. It therefore follows from Theorem 1.6 and Corollary 1.7 that:

1.8 An operator T(E T) admits an extenson T E 1 i and only if the operator w+ (T) has an extension w' (T) E T, and then w+ (T) = w+ (T).

In particular, a (J1, J2)-non-expansive operator (a (J1, J2)-isometric operator) V -admits a (J1, J2)-non-expansive extension (a(J1, J2)-isometric

extension) V if and only if the contraction (isometry) w+(V) admits a contractive (isometric) extension w+(V)E .1, and then j)=w+(V). We now pass on to the description of special extensions of (J1, J2)isometric operators. Definition 1.9:

.2

An extension T of a (J1, J2)-isometric operator V:.'Y1 is called its (J1, J2)-bi-extension if Fv is isotropic in rT relative to the form (x1, x2),
From Definition 1.9 and the fact that graphs of (J1, J2)-non-expansive operators and (J1, J2)-non-contractive operators are semi-definite lineals L]

§1 Potapov-Ginzburg linear fractional transformations

249

relative to the form 2.(4.11), and the graph of a (J1, J2)-isometric operator is a neutral lineal, we obtain, taking 1.Proposition 1.17 into accounts. 1.10 (J1, J2)-non-expansive, (J1, J2)-non-contractive, and, in particular, (J1, J2)-isometric extensions of a (J1, J2 )-isometric operator are (J1, J2)-biextensions of it. The following proposition also follows directly from the Definition 1.9: 1.11

Let T be an extension of a (J1, J2)-isometric operator V If T is a

(J1, J2)-bi-extension of it, then T(r!r n vv,') C j? 1 Conversely, if CA v is a

projectionally complete subspace and T(VT n cV)) C *fr1, then T is a (J1, J2)-bi-extension of the operator V From the Definition 1.9, from 2.Formula (4.14), and from the fact that

the graphs Fv= (<x, Vx) I XE tv) and Fv- = (( V-1y, y) yE liv-t) of the operators V and V -' respectively, regarded as lineals in W1 + M2, coincide, it follows that the following proposition holds: 1.12. Let V be a (J1, J2)-isometric operator with Ker V = (0), and let T be an extension densely defined in .r1 of the operator V Then the operator T is a (J1, J2)-bi-extension of the operator V if and only if V -' C T` and therefore

the operator T` is a (J2, J1)-bi-extension of the operator V -'. The following theorem establishes a connection between (J1, J2)-biextensions of a (J1, J2)-isometric operator and bi-extensions of its PGtransformation. Theorem 1.13: An extension T of a (J1, J2)-isometric operator V is a (J1, J2)-bi-extension of V if and only if w+ (P v) (C .3 (B 04) is isotropic in w+ (Pr) (C.3 O+ 4) relative to the form (X3, y3)3 - (x4, Y4)4. In particular, if T E T, then T is a (J1, J2)-bi-extension of a (J1, J2)- isometric operator V if and only if the operator w+ (T) is a bi-extension of the isometric operator We use the Definition 1.9 and we also note that the left-hand side of the

formula (1.1) coincides with the (- J1,)-metric. Therefore (x1, x2) is an isotropic vector in P r if and only if w+ (x1, x2) = (X3, x4) is an isotropic vector in W' (FT) relative to the form (X3, y3)3 - (X4, y4)4. From this the first assertion of Theorem 1.13 is obtained. The second assertion is a direct consequence of

the first, taking Theorem 1.6 into account.

Corollary 1.7, Proposition 1.8, and Theorem 1.13 show us the way to describe special extensions of (J1, J2 )-isometric operators (see Exercises 5-11). It is well known that in extensions of symmetric operators in a Hilbert space the method of Cayley-Neyman transforms is used, reducing the given problem to the problem of extending isometric operators. In J-spaces such a method is not always applicable, since (cf. 2.Example 3.36) it is not excluded that ap(A) = C. However, in 2.Proposition 2.3 we mentioned another method 4

5 Theory of Extensions of Isometric and Symmetric Operators

250

of extending J-dispative, and therefore also J-symmetric, operators: JA - JA JJA. Finally, in the preceding paragraphs a method of

A

extension by means of PG-transforms has been indicated.

It turns out that the three transformations-the Cayley-Newman transform, multiplication by J, and the PG-transformation-are closely connected.

Theorem 1.14: Let . ' Wbe a J-space and Or =,-W @.,Y be the space of graphs. Then K; 0 [ J, IJ 0 K; ' (x, y) = w+ (x, y) for all (x, y) E Or, here

[J,1]0(u,v) = (Ju,v) ((u,v) E fir).

D K;°[J,I]

°K;1

(x, y)=K;°[J,I] °

x+y x+y 2i

'

2i

-K`Jx+Jy x+y -(x++y-,x +y+)=w+(x,y). 2i

21

From this follows directly.

' be a linear operator: Then: Corollary 1.15: Let T: ,Y( a) TE T, 1 up(T) iOvv(Ki'(T)J); b) TE T, l ap(w+T) = 1 ¢ap(T):

c) 10up(T), i0up (K;'(T)J)- TE T. If the premise of at least one of the conditions a), b), or b) holds, then K;(K; '(T)J) = w+ (T).

§ Exercises and problems 1

Let T:411 ?Pz TP,- _

i Q1 ) .2

z O+ z ), TE .1T, Prove that then V,,-(T) = IY3 (_ i (B -Y2- ), and if T = II T%i ?i=z is the matrix representation of the operator T, then W+(T)1(w+(T))ijl z

w+(T):J''3-y-W4 (= z

D.W

)

has the matrix representation

j=i, where (w+(T))u = T11- T12Tz2'T21, (w+(T))12 = T12Tzz', (w + (T))21 = Tzz' Tz,, (w + (T))22 = Ti-21 (Shmul'yan [5] ). 2

Prove that the PG-transformation w+ establishes a one-to-one correspondence between all (J,, J2)-non-expansive linear relations .4 " C /', O+ .W'2 (i.e., [x2, x2] < [x,, x,] if (x,, xz) E./I") and all contractions V:.W3 -.YP4. Investigate what is the pre-image of isometries, of semi-unitary and of unitary operators (Shmul'yan [5], Ritsner [4]). Hint: Use Formula (1.1).

3

Construct an example of a (J,, J2)-bi-extension of a (J,, J2)-isometric operator which is neither a (J,, J2)-non-expansive operator nor a (J,, J2)-non-contractive operator. Hint: Use Proposition 1.11.

4

Derive formulae connecting the Cayley-Neyman transform Kr for an arbitrary and the Potapov-Ginzburg transformations .., w± (Azizov, E. lokhvidov, and I. lokhvidov [1]. Hint. Use the scheme of the proof of Theorem 1.14.

§1 Potapov-Ginzburg linear fractional transformations 5

6

251

Prove by means of PG-transformations that a continuous, continuously invertible, closed (J1, J2)-isometric operator with finite deficiency-numbers p = -dim 9v 11 dim ,Wv11 admits a (J1, J2)-unitary extension if and only if In 1111 = In i'l 11 (cf. E. Iokhvidov [4], [5] ). Let V be a bounded (J,, J2 )-isometric operator with Vv= 1 v, p = dim 'J

q=dim

X11,

.

r=dim Ker V, m=dim

n=dim

Then

p + q = in + n + r (Azizov, E. Iokhvidov, I. Iokhvidov [ 1 ]; cf. E. Iokhvidov). 7

Give examples of operators for which the numbers p, q, r, in, n in the equality in Example 6 take all the possible values admissible by this equality (Azizov; cf. E. Iokhvidov [4] ).

Hint: Consider a J-space

.

= .YP+ 0,,Y- (where .YP+ = .YY- is an infinite-

dimensional space), and a semi-unitary operator, acting in .h°, U= 11 U;ijj?i=j, where Uii = -K*U2i, KE.X+; U22 = KU12; U21 = S21(S2*IKK*S2i +

1+)-in

is a operator .Yl'+ into semi-unitary carrying .YP-; U12 = S12(S,zK*KS12 + I-)-12, S12 is a semi-unitary operator carrying .YW'- into .Ye+. To solve the problem it suffices to vary the operators K, S12 and S21, and also the number of examples of the space .W, and to put V = w+ (V ). S21

8

Let (m, n) be an arbitrary pair of non-negative integers. Give an example of an unbounded closed (J1, J2)-isometric operator V with

=.;f and m=dim (/+(v), n=dim .Jt4-(/)

6;v

(essentiality of the condition for boundedness of V in Exercise 6). Hint: Use the device given in the hint on Exercise 7. 9

Let V be a (J1, J2)-isometric relation, V1, = 1 v, .;?v = . 'v, and let in, n, p, q, r be

the same as in Exercise 6, and s = dim Ker V -'. Then p + q = in + n + r + s (Azizov, E. lokhvidov and I. lokhvidov [ 1] ). 10

Let .YP be a J-space, T a linear relation, and nr(T) = dim .-IT- rt. Prove that if T is a J-Hermitian linear relation, 1 ;e 3 , and .-1T- tt and WT- r T are subspaces in.YI , then

nl-jT) = nr(JT) + nl-(JT) + dim Ker(T- I) + dim Ker(T- ('I). In particular, if is a point of regular type for T, then the condition that 'WT- r t be closed may be omitted (Azizov, E. Iokhvidov, I. Iokhvidov [1]). Hint: Use the results of Exercises 6 and 9 and also Theorem 1.14. 11

Under the conditions of Exercise 10 let T be a J-Hermitian operator. Prove that T = TT if and only if

n1(T) + nt(T)=dim Ker(T- J)+dim Ker(T- ('I) (Azizov, E. Iokhvodov, I. Iokhvidov [1]). Hint: Use the results of Exercise 10. 12

Prove that

if

T is

a

J-non-expansive operator

and I O ap(T),

then

K;(K;'(T)J)=w+(T).

Hint: Use Exercise 33 on 2.§4 and Corollary 1.15. 13

Let 1=(o[+]12 and 11=('J1)o[+] (1'1) [+] ('J1)2 be the 1 and subspaces

decompositions of the .H = 1'o [+] 1i [+] 12 O+ (11)o [+]

(.YP+,.YP-)-

'j +

let

[+]('/'1)2, and let J= 11 Ullbi=1 be the matrix representation of J (see Exercise 5 on 1.§10). Verify that a bounded

252

5 Theory of Extensions of Isometric and Symmetric Operators operator X with 1'x = .0 represented by the matrix X = 11 X,jII 6;= I will be a J-bi-extension of the operator Iv = I I C1 if and only if 11 X;ijI i;= I = I,, X;j = 0 when i=4,5,6, j=1,2,3; X2;=0 when j=4,5,6; .3 X6jC Rj6b and X31= - J36 FF6' X6i, j = 4,5; 'I6 - x66 Cj66 and X36 = J36 46V - 66);

X44 = I, and all the remaining X;; are arbitrary (Azizov [14] ).

Hint: Write out the matrix X` and use the inclusion If C X' derived from the inclusion in Proposition 1.12. 14

Let .' be a J-space, 9 a subspace of ., and J = II JJlMM ?j= 1 be the matrix representation of an operator J relative to the decomposition .r° = V Q+ V -L. Prove that an operator Y( E :i) will be a J-bi-extension of the operator I, if and only if there is operator an Zzz : V2 - 1-1 such that 00ap(I2 + J22 +(I2- J22)Z22) and Y= II

11

Y1i

%.i= 1, where YI1 = Iv, Y21 = 0,

Y12= -2J12(I2-Z22)(12+ Jzz+(h - J22 V22)

Y22=(I2- Jzz+(II+ Jzz)Zzz)(12+ J22 +(I2- Jzz)Zzz)-' (Azizov [14]). Hint: Verify that w+ (Li) = Ls; use Proposition 1.11 and establish that biextensions of the isometric operator If coincide with the set of operators representable by matrices Z= jI Z;ijI ij= 1, where Z1l = Iv, Z12=0, Z21=0, and Zzz : 9)-L - Y 1; verify that the condition Z E J is equivalent to the condition 00ap(I2+ J22 +(12- J22)Z22) and that Y=w+(Z). 15

Prove that under the conditions of Exercise 14 the operator Y will j-bi-nonexpansive (respectively, J-semi-unitary J-bi-non-expansive, J-unitary) if and only if Z22 is a contractive (respectively, semi-unitary, unitary) operator with 1'z22 = 9 and 0 E p(I2 + J22 + (12 - J22)Z22) (Azizov [14] ). Hint: Use the hint on Exercise 14 and Corollary 1.7.

16

Let V be a (J1, J2)-isometric operator which has a continuous (JI, J2)-biextension Vo with 9 vo = . , a projectionally complete range of values, and Ker Vo = 161. Prove that the operator P with VV= W 1 is a (JI, J2)-bi-extension of

the operator V if and only if V=I.,,Vo(VCVo)-', where I.,r is a certain J2-biextension of the operator I.,? = 12 v6; V'i.,,. = y f2 (Azizov [ 14] ). Hint: Use the fact that Vo Vo is a linear homeomorphism of the space .Yi and is a J1-bi-extension of the operator I I Vv. 17

§2

Prove that if, under the conditions of Exercise 16, the operator Vo is (JI, J2)-semiunitary, then the formula given there can be rewritten in the form V = I.,QVo, and if Vo is a (JI, Jz)-unitary operator, then V is a (J1, J2)-unitary (respectively, (JI, Jz)-semi-unitary, (JI, J2)-bi-non-expansive, (JI, Jz)-bi-non-contractive) operator if and only if i.w belongs to the corresponding class (Azizov [14]).

Extensions of standard isometric and symmetric operators

; _ .YP; Q+ W1 be Ji-spaces (i = 1, 2). In this section we present 1 Let another approach to the construction of a theory of extensions of (JI, Jz)isometric and J-symmetric operators and we describe all their (J1, J2)-biextensions and J-bi-extentions respectively.

§2 Extensions of standard isometric and symmetric operators Definition 2.1: We shall call an operator V: .W'1

253

W2 a standard operator

and write V E St(."1, ."2) if V is a closed continuous and continuously invertible (J1, J2)-isometric operator. In particular, if ._W, = .W'2 = ,Y is a J-space, we shall write V E St X. We point out that the condition VE St(, '1, 72) implies that Yv and Rv are closed. Using Zorn's lemma it is easy to prove that every (J1, J2)-isometric operator

V admits extension into a maximal (J1, J2)-isometric operator V, i.e., V now has no non-trivial J1, J2)-isometric extensions. Less obvious is the assertion that if V E St(7,, M2), then among its maximal extensions there is a V E St(7?'i, M2). Theorem 2.2 is devoted to the investigation of this question; in the proof of this theorem a construction for the corresponding extension is given which will be used more than once later.

Theorem 2.2: Let VE St(1,.2). Then it has a maximal (J1, J2)-isometric extension VE St(,W1, '2).

First of all we observe that if 9I-1 and RI1l are regular subspaces, then, as is easily verified, the operator V has an extension V'E St(,,, .Y2) such that its deficiency subspaces 9 vl] and ,3W are regular and either at least one of them is (B) or they are uniformly definite and have different signs. The maximality of such (J1, J2)-isometric operators is obvious. Consequently the theorem will be proved if we verify that every operator VE St(,,,M2) admits extension into an operator V' E St(.7,, W2) with regular deficiency subspaces VVJ and We consider the (,7i , W1 )-decomposition of the subspace 9v (see 1.Definition 10.3): RV).

V V = C/o [-H 91 [+] ci2

(2.1)

and the (.7?', .z )-decomposition of the subspace Rv:

Rv=Ro[+] W, [+]12.

(2.2)

Without loss of generality we shall suppose that (2.1) and (2.2) are connected

by the relations VCA, = R,, i = 0, 1, 2. Indeed, Ro = V90, since a (J1, J2)isometric operator maps the isotropic part go of the subspace Vv on to the

isotropic part Ro of the subspace Rv. But if even one of the relations R, = V '1 (i = 1, 2) is infringed, we proceed as follows. Taking 2.Corollary into account we shall suppose, without loss of generality, that v((' 1 n .-i) C .Y( and we consider the decomposition

4.13

Jv= Vo [+] V-'R1 [+] V-'R2.

(2.3)

Hence 2= [4-J ]J , (1 C V-'R1, V-1R1 = c1 [+] V2, and where 92"= V-'R2 By virtue of 1.Theorem 10.4 we can find a maximal 11

5 Theory of Extensions of Isometric and Symmetric Operators

254

uniformly definite dual pair (Y1', Yi) such that (2.3) will be the (2l+, Y; )-decomposition of the subspace 9v. It only remains to put Let

Q'V =(V1)0[+]W)I[+]W)2 and

YV = (

1)0 [+] (

1)1 [+] (R1)2

be the (771 , .W )-decomposition and the (.,Yz , .,Yz )-decomposition of subspaces V 'V and i.Rt respectively. Then relative to the decomposition

the

.Iy1 = /0 [+] t1 [+] 92 O (V')o [+]W)1 [+](d 1)2 the (respectively, '2 = ,!0 [+]4?1 [+] 2 O 0?1)0 [+] 0?l)1 [+] (.1)2) operator J1 (respectively, J2) can be expressed in accordance with Exercise 5 on 1.§10 in the form of a matrix J1 = I Jtil) I61 (respectively, J2 = 11 JJ(jzt1I -=1.

The components of the expansions of the matrices J1 and J2 have the properties indicated in Exercise 5 on 1.§10. Thus, for example, J16) _ 3>2)1i2V36),

(131) - A3>2)"2 06), and j162) = (I 2) - J

where V}6) is an isometry

mapping (d 1)2 on to d2, and V36) is an isometry mapping (1)2 on to JV2. We introduce the notation V;+1 = V I d; (i = 0, 1, 2) and define an extension V' of the operator V in the following way

V'Idv=V; V' X1)0

V4:(V1)0- (1)0,

V4= Jgl)V

1J14);

2 0 W)2, V6X= V36X+ V6X, XE ((1)2,

V' I(d1)2 = V6: (d1)2 V36X E 32, V6X E (3q1)2,

V36 = [ (132) + (J2) -

J33)2)1/2)- I J33

V*- I

(Jill - J33)2)-1)J33)] 116), V6 = V36)* V3 *-1 V36).

The operator V' is continuous and continuously invertible; (I V' = V O [+] (I1 [+] V2 @ (V 1)0 [+] (V' )2, and

RV'=,3? [+]31[+]+J?2(@ ( 1)0[+](91)2.

Moreover, V'(do 0 (V1)0) = ,Ro O ( 1)0, V' d1 = ill, and V'((I2 O (d1)2) = W2 +O (-?')2. In accordance with Exercise 4 on 1.§10 the subspaces and `do O+ (d 1)o, d 1, and 92 O+ (V')2 (respectively, 4?o O+ ( 1)0, 1,

.R2 O+ (1)2) are pairwise JI-orthogonal (respectively, J2-orthogonal), and these subspaces are invariant relative to J1 (respectively J2). Therefore V' will be a (J1, J2)-isometric operator if and only if the operators V' `/b O+ (V 1)0, V' (I 1, and V' I V2 O+ (I 1)2 are (J1, J2)-isometric. Since V' V I = V I (I1, the operator V' (l is (J1, J2)-isometric (by hypothesis). Let XI E ('o, I

§2 Extensions of standard isometric and symmetric operators

255

X4 E (`I' 1)0. Then V' (x1 + X4) = V1 X1 + J4f) V,'-' Ji4')x4, and

J2V'(x1+x4)= J13VIx1+ Vl* 1Ji4X4. Consequently 1

[V'(X1 + X4), V'(XI + X4)12 = (Jf4)*Xl, x4)1 + (J14)x4, x1)1 = [xl + X4, X1 + X41111,

i.e., V' I 1o ®+ ('1)o is a (J1, J2)-isometric operator. We verify that the operator V' I V'2 O W)2 is also (J1, J2)-isometric. To do this we shall suppose, without loss of generality, that W1 = V2 O+ (91)2, %P2 = R O+ (,W1)2, and we shall prove that V' is a (J1, J2)-unitary operator, i.e., V'* J2 V' = J1. It can be verified immediately that the operator T = V' * J2 V' has the matrix representation

I

T33

T36

T63

T66

where T33 = V3 Jf3' V3,

T63 = T36 = V3 J33 V36 + V3(I32) - J33

2)1/2 I/6,

and T66 = V36 J33) (V36 + V6 * V36 *(I32) - J33)2)1/2 v3(2 + V36 (I32)

6

J(3)2)1/2 "36

v,6 -

V6 *

V3(6)* j3(2) 3

32

V2

V6.

Since the operator V3 is (JI, J2) isometric, we obtain T33 = J431). We substitute the values of the operators V36 and V6 in the expressions for T36 and T66, and after some elementary transformations we find that T36 = Jf b) and T66 = 46), i.e., T= J1. Thus V' I 942 O+ (ft')2 is a (Ji, J2) isometric operator and

V' E St(.01,.W'2). Moreover, (y1)1 and (.R1)I are its deficiency (regular)

subspaces and therefore V', and with it also V, has among its maximal (J1, J2)-isometric extensions also operators VE St(.'1,.-W2).

For an operator VE St(.Iy1,.2) to be a maximal (J1, J2)isometric operator it is necessary and sufficient that at least one of the Corollary 2.3:

following conditions shall hold: a) VVl] = )B), i.e., V is a (Jr, J2)-semi-unitary operator;

b) .1111 = )B), i.e., V-' is a (JI, J2)-semi-unitary operator, c) VI1J and R11] are uniformaly definite subspaces of different signs.

The sufficiency of each of the conditions a)-c) for maximality of a (J1, J2)-isometric operator V is obvious. Now let V E St(.1P1i .2) be a maximal (J1, J2)-isometric operator. From the course of the proof of Theorem 2.2 it follows that #l and ow;,-'l are regular

256

5 Theory of Extensions of Isometric and Symmetric Operators

subspaces. Let us suppose that none of the conditions a)-c) is satisfied. Then there are B ? xo E and 9 * yo E t such that [xo, xo] = [yo, yol. We define an operator V' : Lin (V v, xo) - Lin) i v, yo) , V C V', V' xo = yo. It is easy to see that V' E St(.°1i '2)-and we have obtained a contradiction. J

Corollary 2.4: Every operator V E St(. 1, .W'2) admits a (JI, J2)-bi-extension Vo such that :91 vo is a projectionally complete subspace and at least one of the operators Vo or Vo has a trivial kernel.

By virtue of Theorem 2.2 we can suppose that VE St(. '1i'2) is a maximal (JI, J2)-isometric operator. We now use Corollary 2.3 and Proposition 1.11. If V satisfies condition a), then we put Vo = V; if V satisfies

condition b), then we put Vo = (V -' )`; if V satisfies condition c) and dim qV] < dim 5t1then as Vo we take any extension of the operator V on to WI which maps

I, then as Vo we take any extension of the operator Von

to WI which mapsY!A'I injectively on to a subspace in RV'I, and if dim IL] > dim 9l `l] , then we put Vo = (V-' )`, where V-' is an arbitrary extension of the operator V subspace in ILl

on to W2 which maps 9 I'l injectively on to a

Thus, for the description of all (JI, J2)-bi-extensions of an operator VE St(.1,.2) the results of Exercises 13-17 on §1 can be used.

2 We pass on now to study the problem of the possibility of (JI, J2) unitary extensions of operators VE St(. 1, JY2). Example 2.8 given below of an operator V E St(.W1, ."2) with 9v = Rv (and therefore 9 fr1 = 3 *) shows that, in contrast to the Hilbert space case (see, e.g., [I] ), in the case of a Pontryagin space (see Exercise 3 below) information only about the properties of 9 Vll and f'l is not always sufficient to make conclusions about the possibility of extending an operator V into a (J1, J2)-unitary operator. But first we introduce some preliminary materials. Definition 2.5: Let (91+, 2-) be a maximal uniformly definite dual pair in a J-space . , and suppose the subspaces A"+ E -ff+ and ./I-- =.4'[+-,I (E 11-)

contain no infinite-dimensional uniformly definite subspaces. Then the number v (.,l "+) = dim(./l "+ n Y+) - dim(,il n Y-) is called the index of the subspace . 4'+.

Lemma 2.6: In Definition 2.5 v (. 'l '+) does not depend on the choice of the (Y+, Y-). maximal uniformly definite dual pair Let

P1( l"+)=dim(.,l"+ fl.,Y+)-dim(./l"_ n.w- )

§2 Extensions of standard isometric and symmetric operators

257

and

v2(. '+) =

dim(.'{"+ n q+)

- dim(.'1i. fl g'-).

We verify that v, (,/1'+ ) = P2(,4'+ ).

Let K+ be the angular operators of the subspaces A-+ and Q± be the angular operators of the subspaces 9?± . From Exercise 17 on 1.§8 and Exercise 13 on 2.§2 it follows that K+ is a 4)-operator and I K+ I = I+ + S, where S E Y.. From Exercise 14 on 2. §2 it therefore follows that K+ - Q+ is also a 4)-operator and ind K+ = ind(K+ - Q+). It remains to observe that ,4'+ n ± = Ker K+, A"± fl 2± = Ker(K+ -Q±), and K+=K*, K+-Q+=K*-Q*, i.e., -ind K+ and v2 (-4"+) = - ind(K+ - Q+ ).

Lemma 2.7: If VE St(.Y(1,.Y2), d'vE M+(JY1), .9lvE,11+(."2), and 1'v, Rv, gW and 91VI contain no infinite-dimensional uniformly definite subspaces, then V admits a (J1, J2)-unitary extension if and only if v(Vv) = v (91 v ). Moreover every maximal extension V' E St(.Y1, '2) of the operator V will be (J1, J2)-unitary. Let 17 be a (J1, J2)-unitary extension of the operator V. We introduce in

.Y'2 a scalar product (x, y)2' = (V-'x, V-'y)i (x, yE.)2) equivalent to the original one. Since [x, y]2 = (Jzx, y)2, where Ji = 17J1V-' = Jr', the components of the new canonical decomposition 2 = 'Y2" O+ .VZ 'will be the subspaces ,Yf F.01'. Consequently dim(vv n i ) = dim(V(cv n .)Yi ) = dim(3 v n ,y2,,) and dim(91*1 fl e-) = dim(V(Qo ] fl Yi 1)) _ dim( V, fl W '), and therefore v(V v) = v(,R v). Now let P(Vv) = v(Rv) Without loss of generality we shall suppose (see the proof of Theorem 2.2) that the decompositions (2.1) and (2.2) of the subspaces cv and Rv are connected by the relations VCA; = Ri, i = 0, 1, 2.

Since c1 = Iv n .iYi and 1 = 91v fl .w2+, we have dim(iv n i) = dim(3ly n . w2+). Consequently dim(cv, n W,-) = dim(f1w fl. l-) = -v(/v)+dim(Vvn -wi v(JRv)+dim(.91vn,Yzdim(.RVI n.YZ ) = dim(MV n . i ).

Following the scheme of the proof of Theorem 2.2 we deduce that the operator V admits an extension V' E St( 1, . 2) with 9W n .01- and wv n Wi . Hence V VJ and [1I are negative subspaces of equal finite dimension and therefore it is now easy to construct a (J1, J2)-unitary extension of the operator V' and it will also be an extension of V. Let V' E St(.,*P1, -02) be an arbitrary maximal (J1, J2)-isometric extension

of the operator V, and suppose that V' is not a (J1, J2)-unitary operator. Since V;,1l and ,j-] have the same sign, we conclude from Corollary 2.3 that we can, without loss of generality, suppose the operator V' to be (J1, J2)-semiunitary and dim R Vi] ;4 0.

We consider the J2-space ,Y2= V'. 1. From what has been proved, P(Vv) = v'('), where P'(') is the index of Mv in .W''. But by hypothesis

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5 Theory of Extensions of Isometric and Symmetric Operators

and it is easy to see that v(3'v) = P' (3 ) - dim .91(1j-so we v(V v) = have obtained a contradiction. We pass on now to the construction of an operator VE St Y' with (v= ,3v, but which does not have J-unitary extensions.

Example 2.8: Let V1, LA2, (!Yl)2 be infinite-dimensional separable Hilbert spaces. We form the space ,Y = 91 Q+ CA2 Q+ (GO ')2 and introduce in it a J-metric (see 1.Example 3.9) by means of the operator

J=

J22

0

0

J33

0

V36 (I3 - J33)

0

(J3 - J33 1/2

)1/2

V36

,

- V*J03 V36

constructed relative to this decomposition of the space ye, where J22 = 12, J33 E Y., 0 < J33 < 13, and V36 is an isometry mapping ('W1)2 on to 92. Let be the eigenvalues of the operator J33, and let X1 >, X2 >, limk-., Xk+1/Xk = a > 0; let [ fk) be an orthonormalized basis in 92 composed of the eigenvectors of the operator J33: J33fk = Xkfk (k = 1, 2, . . .); and let [ ek 11 be an orthonormalized basis in 91. We define on 9 v = 91 O+ 92 an operator V E St W by putting V e k = ek+1 (k = 1 , 2, .

.

.),

Vf1= Xl el,

V f k = Xk+1/Xkfk

(k = 1, 2, ...).

By construction My = cv. The operator V admits no J-unitary extensions V, for if it did the operator VI V2 (E St(,'1, ,1W2)) acting from the J1-space .)r1 = C42 O (4 i )2 into the J2-space W2 = Lin [ el) + CA2 O+ W)2 with Ji = J I Yei, i = 1, 2, would have (J1, J2)-unitary extensions, and this is impossible by virtue of Lemma 2.7 since P(92)= 0 and v(Vi2) = 1.

The above example shows that in answering the question whether an operator VE St(.'1,,W2) has (J1, J2)-unitary extensions one has to take into account not only the characteristics of the spaces CA I1l and RI'l but also the `action' of the operator V. Let be a maximal uniformly definite dual pair in Ye1, let V V = C/o [+] 1/ 1 [+] V2 be the ('l', 21 )-decomposition of the domain of definition of an operator V E St(,*'1 i 46), and 91 v = 3o [+] V!Y 1 [+] Vc12 be the decomposition of its range of values. We choose in .-W2 a maximal uniformly

definite dual pair (Y2, Yz ) such that Vw v fl 99i ) C Y27+, and we introduce the following constants v+ (V) and v _ (V ): if there is in V[,1l or in an infinite-dimensional uniformly negative (respectively, uniformly positive) subspace; (2.4) dim(V I-L] fl Y ) + dim( V/2 fl £Z) - dim(. 1[,1] fl Pz ) otherwise. 0

v±(V) =

§2 Extensions of standard isometric and symmetric operators

259

Remark 2.9: If there are no infinite-dimensional uniformly definite subspaces in Vv, v, 9111 and R(,1], and if 9v E,.tl+(.W'1), and i?vE tl+(,°z), then

v_(V)=0 and v+ (V) = v(Av)- v(C'v). Our immediate purpose will be to prove that v±( V) are independent of the choice of the dual pairs (-Ti', 2 ), i = 1,2. We preface this with the following proposition.

Lemma 2.10 Let . be a J-space, f a subspace of it, and let U be a J-semi-unitary J-bi-non-contractive (respectively, J-bi-non-expansive) extension ;e (0) implies that there is in c(1) of the operator I I V. Then the condition an infinite-dimensional uniformly negative (respectively, uniformly positive) subspace.

Let !J = go [+] c1 [+] 92 be the )-decomposition of the subspace J, and let U be a J-semi-unitary J-bi-non-contractive operator. Since 90 and Col] are invariant relative to the operators U and U`, it follows in accordance

with Exercise 20 on 2.§4 that the operator U induces in the i-space _ Col]

!I o

a

J-bi-non-contractive

Jsemi-unitary operator

U1 9 = J1 9, where I = 9/ go. Moreover, ML'I ;;d

(0)

U,

and

if and only if

U ;e (0); and 9[1] contains an infinite-dimensional uniformly negative subspace if and only if (1] _ 9 [1]' go has this property. Therefore we shall suppose without loss of generality that go = (0), i.e., ', and with it also iI [1] and g', are non-degenerate subspaces. In accordance with 2.Proposition 4.14 we have for the J-bi-non-contractive operator U that M I l (= Ker U`) is a uniformly negative subspace, and so without loss of generality we shall suppose that 91Il C -W-. We shall assume

that W # (0) and that, contrary to the proposition, V[l] does not contain infinite-dimensional uniformly negative subspaces, i.e., (see Exercise 7 on 1.§6) the Gram operator of every non-positive subspace from c l1] is completely continuous. Let U = II U jII ? j=1, J = II J;i ?i= 1 be the matrix representations

of the operators U and J relative to the decomposition W = SI (1 91 . Since mil[ 1 C Y and ci CRC, we have J& C V`(= Jc [1] ), and U I = 9lu2, n ci 1 = i i I n c 1. By direct calculations taking Exercise 7

on 1.§3 into account we verify that the fact of the operator U being J-semi-unitary is equivalent to the relations a) .V1, - u22 C . IrZZ, U12 = - J21 Jzz' (Iz - U22); b) Uzz Jzz' Uzz Jzz = Iz.

Since Ker Uzz =

u,z n `I' 1, and condition b) is the condition for the operator be to Jzz-semi-unitary, so

( Uzz) ` = Jzz' Uzz Jzz

(Ker U2*2)111). Let CA1 = c+ [+] V_' be the canonical decomposition of 'J 1, and let P+' be the projectors on to V +, P+ + P_ = 12. It follows from the relation a) and the complete continuity of J2'21 Y_ that the operator

T= P_Jzz'(I2- U22 )J221 ci_' =I_ - P_Jzz'UzzJzzI f'

260

5 Theory of Extensions of Isometric and Symmetric Operators

is completely continuous, and therefore P_ Jzz' U22 J22 9' = I_ - T. Since Jzz' U22 J22 is a J22-semi-unitary operator, we have Ker(P_ Jzz' U22 J221 CA') _ (0) and therefore i?j - r= LA-. Hence we conclude that A2' U22 J22 C4' is the

maximal non-negative subspace in 9 1. But the subspace Ker Uzz (6 {6) ), which is J-orthogonal to ,?(uiz)', is also negative-and we have obtained a contradiction. The case when U is a bi-non-expansive operator is proved similarly. Lemma 2.11:

In Definition (2.4) v±(V) do not depend on the choice of the

dual pairs (?; , 2' ), i = 1, 2. We shall verify that v+( V) does not depend on the choice of the dual pairs the argument for v- (V) is similar. If there is an infinite-

dimensional uniformly negative subspace in mfr], then v+(V)=0 by definition does not depend on the choice of the dual pairs. Suppose all the uniformly negative subspaces in Vl] are finite-dimensional. Without loss of generality we shall assume that the decompositions (2.1) and

(2.2) are connected by the relations Vii; = 3;, i = 0, 1, 2, and therefore the

constant p+(V)v, calculated with respect to the dual pairs i = 1, 2, coincides with the difference dim(9f1' fl;)-dim(3[1I fl .z). We bring into consideration a J,-space ;, and a J2-space 46, putting , y , [ O+ ].

1,011

0]0 1,0-

,W2[O] I

where dim

-_I

P, I,

j, IYe

G)

_

±K± if v, <0,

i2I,'Y±

[OJ

+, Ye2[O+],W+,

and ,Y+

if v, > 0,

J, .W+ = I+

+,

Jzl ye+I+ is

=±1±

if v,>0, if v, 50,

an infinite-dimensional space with

dim + > max(dim ,, dim .)Y21. It is clear that the constant v+(V) = v, (see (2.4)) constructed relative to 4e ,t, . e, ), i = 1, 2, is equal to zero, and it is

sufficient for us to verify that it does not change for any maximal uniformly definite dual pairs (S,+, i = 1, 2 containing (.YP+, .Yfo ), where .moo = if Y- enters into the considered space ; , (i = 1, 2), and No = (0) otherwise. Since J, = 0, we can show, by repeating the arguments used in the proof of Theorem 2.2, that the operator V has a (J,,12 )-unitary extension V,. Let JZ be 7), the constant (2.4) calculated relative to some other of the dual pairs i = 1, 2, indicated above. Again reverting to the proof of Theorem 2.2 and

using Lemma 2.7 it can be proved that the operator V has a maximal (J,, j2)-isometric extension V2 E St(.;,,.ie2), and that one of the operators V2 or VZ ' is defined on the whole space, and the orthogonal complement to its range of values is negative and has dimension I v2 1. Let V2 be such an operator (for Vi- ' the argument is similar). Then the operator V,- 'V2 is a j,-semi-unitary J,-bi-non-contractive extension of the operator I I i v and 9 [ J contains no infinite-dimensional uniformly negative subspaces. By virtue of Lemma 2.10

§2 Extensions of standard isometric and symmetric operators

261

V,-'V2 is a Jl-unitary operator, and therefore V, is a (J,, j2)-unitary operator, which implies the equality j72 = 0.

We now pass on to the main result of this section. Theorem 2.12:

An operator VE St(.°,, 42) admits a (J,, J2)-unitary exten-

sion V if and only if v+ (V) = v _ (V) = 0 and there is an operator V, E St(.',, J '2) which maps !2V1' on to &V1I

Let V be a (J,, J2)-unitary extension of the operator V. We then put V 9 V = V, E St(°,, . 2) and it is clear that V, 9 Vl' _ M Vil. The equality v+ (V) = v_ (V) = 0 follows from Lemma 2.11 since this equality holds for the

dual pairs (.01', ,i) and (V1+ , V.Wj ). The converse assertion will be proved in several stages. a) Suppose first that v+ ( V) = v _ (V) = 0 and that 9 VI and V1' contain no infinite-dimensional uniformly definite subspaces. Repeating the arguments

used in the proof of Theorem 2.2 we obtain that the operator V admits (J,, J2)-unitary extensions. b) Let V1' contain infinite-dimensional uniformly definite subspaces of both signs, let V, E St(.W,, 02), V,9V1' = M V ], and let V'(E St(JY,,.'2) be a maximal (J,, J2)-isometric extension of the operator V. We go into the case when V' is a (J,, J2)-semi-unitary (J,, J2)-bi-non-contractive operator and we show that it can be `touched up' into a (J,, J2 )-unitary extension V of the

operator V. The remaining cases are verified (taking Corollary 2.3 into account) by the successive application of a similar procedure. Since 9 Vi] contains infinite-dimensional uniformly negative subspaces, MI'I (= V, 9 Vl' ) also has this same property. Let q2 be such a subspace 2', = Vi '22. Since let from 9LVlI R[1] and containing

and 9?, and V' 2'1 [+] M V,I are uniformly negative subspaces, there is an operator V(E St(.Y,, '2) which maps 2', on to V' 2', [ f ] M VI. It only remains to put dim 9, = dim( V' 2', [+] 91 V1 ),

(V'x,

Vx= (V, x,

xE 2'1'] X E 9?,

c) Now suppose that V V I contains infinite-dimensional uniformly definite subspaces of one sign and only finitely-dimensional ones of the other sign, and v+ (V) = v_(V) = 0. In this case a combination of the arguments of stages a) and b) is applied. Remark 2.13: In stage a) in the proof of the converse assertion in Theorem 2.12 the existence of the operator V, was not used, and in stages b) and c) in

this proof the operator V, was used only in the search in ( VI'' for an infinite-dimensional uniformly definite subspace of the same dimension as the corresponding subspace chosen in J?VrI

262

5 Theory of Extensions of Isometric and Symmetric Operators

If under the conditions of Theorem 2.12 91V1] = Vo [+] CA [11 where '/b = vv n V v`], and t [1[ ' is a projectionally complete subspace in .YP1, then the existence of a (J,, J2)-unitary extension V is equivalent to the In equality In Corollary 2.14:

El It follows from the conditions on 91 ] that 9 v = Vo [+] 9', where V' is a projectionally complete subspace (see Exercise 7 on 1.§7), and therefore ?v and have property. a (_ .Ro similar Let v1[

(`e[1[)+ [4](t'f'-))_ and W[-Ll' = (,W[-LI)+ [+]By hypothesis , and (,[1[)+, (.i?[1) + are

dim V'o = dim 3 o, dim(V[1[)± = dim(M [1) +

uniformly definite subspaces, and so there is an operator V, E St(.,, .02) mapping VI'] on to V Taking as corresponding maximal dual pairs q ), i = 1, 2 the pairs of spaces 'i 3 (V [1) ± , 2'z 3 (.i [1]) ± we obtain that v+ (V) = v _ (V) = 0. It only remains to apply Theorem 2.12. The converse assertion is obvious.

3 We introduce the concept of x-regular extensions of operators VE St(. ,,.-N'2), and we give a criterion for their existence. But first we prove the following simple proposition.

Suppose that P is a (J,, J2)-non-expansive extension or a (J,, J2)-noncontractive extension of an operator V E St(.W,, W2) on to the whole of .01, that V admits a (J,, J2)-polar decomposition V = WR, where the operator W is a (J,, J2)-semi-unitary operator or is conjugate to a (J,, J2)-semi-unitary operator, and the J,-selfadjoint operator R satisfies the conditions R2 = V`V and a(R) C [0, co). Then W is an extension of the operator V Let x E V v. Then R 2x = 17']7x = x, and since - 1 E p (R) we also have Rx = x. Consequently W 3 V.

2.15

Remark 2.16: If under the conditions of Proposition 2.15 the operator W is (J,, J2)-semi-unitary, then it is a (J,, J2)-bi-extension of an operator V (see

Proposition 1.10). We note also that if V is a J,, J2)-bi-non-expansive operator, then by virtue of 4.Theorem 1.12 it admits the decomposition indicated in Proposition 2.15, and moreover W is a (J,, Jz)-bi-non-expansive operator and therefore (see Proposition 1.10) W is a (J,, J2)-bi-extension of the operator V.

Definition 2.17: We shall say that an operator V E St(.",, .W2) admits x-regular extensions (and when x = 0, regular extensions) if there is a Pontryagin 7r,-space II, with x (< oo) negative squares such that the operator V admits (J,, J2)-unitary extensions V: 4i @+ fI - .)Y2 0+ II,,,

(here 7r = J in the space II).

J1 = J1 Q+ 7rx,

J2 = J2 (+ 7r

§2 Extensions of standard isometric and symmetric operators

263

Theorem 2.18: Let v fl St(.r,,.2 ). Then there is a J-space.r such that the operator V admits a (J,, J2)-unitary extension

V: .r, O+ Jr ' r2 (@ . i,

Jr = J; (D J,

i = 1, 2.

Moreover the following conditions are equivalent: the operator V admits a) regular extensions; b) x-regular extensions; c) (J,, J2)-bi-non-expansive extensions.

We shall suppose that VE St(.0,,.r2) is a maximal (J,, J2)-isometric operator. From Corollary 2.3 its deficiency subspaces and W-'] are projectionally complete. To prove the first assertion it is sufficient to choose as Jr the J-space .-W® D Jr - with infinite-dimensional components Jr

dim Jr ± > max (dim 001'1, dim 11 V] }, and to use 2.14.

We pass on to the proof of the equivalence of the conditions a), b), c).

a) - b) This is trivial.

b)-c) Let V:.r, O+ fl -.r2 H, be a x-regular extension of the operator V, let IIx = TI+ O+ II-, dim II_ = x < co, and let P; be the J;-orthoprojectors from i on to the J space .r, = *, O+ II_, J; = J; O+ - I_, i = 1, 2. It follows from Exercise 34 on 2.§4 that P2VPI I.h1j is a (Fi, J2)-bi-nonexpansive operator, and since Vv C i, R v C .Vz, it is an extension of the operator V. We make use of Remark 2.16 and suppose that, for example, the operator V- 1 has a (J2, Ji )-semi-unitary (J2, JI)-bi-non-expansive extension V' (if V has a Ji, J2)-semi-unitary (JI, J2)-bi-non-expansive extension, then

the argument is similar). Let V' = Vij' - --, be its matrix representation relative to the decomposition .Z = .'2 U H_, Jr( = . , O+ U _ , and let x+ = dim MV1'. Then the operator V' = I Ker Vz, is a (J,, J2)-isometric exten-

sion of the operator V -', and since dim Jz Ker V21 < co, so V v and v, decompose into the direct sum of isotropic and projectionally complete subspaces. Let In 1 (xo, x+, x-' ). Then In

_(xo,x_+x+,x_

Since II _ C V ,] and II _ C. of ,] , we have In(

[,'flr2)=(xo,x+, x_ -x)

and

-x), and moreover

dim(co[,1'fl.W2)=xo+x++x_ -x < 00

.

Now, carrying out an argument similar to that used in the proof of Corollary 2.14 we obtain that V -' has a (J2, J, )-semi-unitary (J,, J,)-bi-non-expansive extension, the (J2, J, )-conjugate to which will indeed by the required extension of the operator V.

264

5 Theory of Extensions of Isometric and Symmetric Operators

c) - a) We again make use of Remark 2.16 and we shall suppose that, for

example, the operator V-' has a (J2, Ji)-bi-non-expansive (J2, Jl)-semiunitary extension W. Since (see 2. Remark 4.29 and 2. Proposition 4.14) 5$wl is a uniformly positive subspace, so by virtue of Corollary 2.14 the operator W has a (J1, J2)-unitary extension W : ,7 '2 O+ . Je1 O+ .', J; = J1 O+ I, i = 1.2, where , is an infinite-dimensional space with dim .c > dim 91wl. Con-

sequently W-1 is a regular extension of the operator V.

In conclusion we introduce some definitions concerning J-Hermitian operators, and we formulate in Exercises 6-13 the corresponding results 4

obtained by applying as above the Cayley-Neyman transformation. Definition 2.19: We call an operator B a J-bi-extension of a J-Hermitian operator A if A C B and the graph I'A is isotropic in I'B relative to the form

[(x1, x2), (Y1,Y2)]r= i([x1,Y21 - [x2,Y1] (2.Formula (1.4)).

The set of closed J-Hermitian operators A with points of regular type (Xo, Xo) (Xo 5;d ko) will be called Xo-standard and will be denoted by the symbol St(.h"; Xo) (= St(Jf; ko)).

Definition 2.20: We shall say that an operator A E St(,'; 4) admits a x-regular (0 < x < cc) Xo-standard extension if there is a Pontryagin ax-space II with x negative squares such that in the J-space = W C+ II

(J = J @ H.) a J-Hermitian operator A has a J-self-adjoint extension A E St(,*; Xo). When x = 0 such an extension will be called a regular extension.

Exercises and problems 1

Let V E St(.r#1, k2). Prove that v, (V-1) = - v, (V) (Azizov). Hint: Use Formula (2.4) and Lemma 2.11.

2

Suppose that VE St(./P1, IY2), that V has at least one (J1, J2)-unitary extension, and that ciH and RI-LI contain no infinite-dimensional uniformly definite subspaces. Prove that then every maximal (J1, J2)-isometric extension V' (E St(. 1, .,Y2 )) of the operator V is (J1, J2 )-unitary (Azizov). Hint: Prove that x ± (V') = 0 for any extension VE St(.10'1,./<'2) of the operator V.

3

Prove that V admits a (J1, J2)-unitary extension if and only Let V if XI = x2 and dim V47L] = dim . H (cf. [XIV]). Hint: Use I. Theorem 9.11 and Corollary 2.14.

4

Prove that under the conditions of Corollary 2.14 the projectional completeness of

v'lll' is essential. Hint: Use Example 2.8. 5

Let A be a J-Hermitian operator, B an extension of it, and VB = ./l'. Prove that B will be a J-bi-extension of the operator A if and only if A C B. Hint: Use 2. Proposition 1.3.

§2 Extensions of standard isometric and symmetric operators 6

265

Let A be a J-Hermitian operator, B an extension of it, k ;4 X, and k ¢ a,(B).

Prove that B is

a

J-bi-extension of the operator A if and only if

its

Cayley-Neyman transform Kx(B) is a J-bi-extension of the J-isometric operator K),(A).

Hint: Use the definitions of J-bi-extensions of J-symmetric and J-isometric operators, and also 2.Formulae (6.29) and (6.31). 7

Prove that A E St(.Y; ko) if and only if ko 0 ao(A) and K0(A) E St W. Hint: Use 2.Propositionl6.11.1

8

Let A E St(.W; ko). Prove that among its maximal J-symmetric extensions there are operators A E St(.YY; ko) (Azizov).

Hint: Combine the results of Exercises 6, 7, Theorem 2.2 and Corollary 2.3. 9

Prove that a J-symmetric operator A E St(.W'; ko) will be a maximal J-symmetric operator if and only if at least one of the following conditions holds: a) ko E p(A); b) ko E p(A); RA - aot and A- a.t are uniformly definite subspaces of different signs (Azizov [14]).

Hint: Use Corollary 2.3, having first verified that a J-symmetric operator A E St(.YP; ko) is maximal if and only if the J-isometric operator K>,,,(A) E St ,Y is maximal. 10

Prove that every J-symmetric operator A E St(.Y'; ko) admits J-bi-extensions Ao such that one of the points Xo or Xo is regular for them, and the other is a point of

regular type and .RA - k,i,

A - ),,,i

are projectionally complete subspaces

(Azizov).

Hint: Use the result of Exercise 7 and Corollary 2.4. 11

Suppose that A E St(.W; ko) and that it has a maximal J-dissipative extension Ao with Xo E p(Ao). Prove that the operator A has maximal J-dissipative extensions A E St(.YP; ko) with ko E p (A) (Azizov [ 14] ).

Hint: Use the result of Exercise 6, Remark 2.16, Proposition 2.15, and 2.Proposition 6.11. If necessary consider 2. Remark 4.29. 12

Let A and AO be the same as in Exercise 10, ko E p(Ao), Im ko < 0, and let Uo = Kx0(Ao). Prove that the values of the resolvents Ra0(A) of all J-bi-extensions

A of the operator A with ko E p(A) are described at the point ko by the formula R4(A) = (I>,0Uo(UoUo)-` - I)12i Im Xo,

where lx traverses the set of all J-bi-extensions of the operator I I WA - 41; if, in addition, Ao E St(.W; ko), then R,,0(A) _

I)12i Im Xo,

and when AO = Ao the operator A will be J-symmetric (respectively, maximal J-dissipative, J-self-adjoint) if and only if ix, is a J-semi-unitary (respectively, J-bi-non-expansive, J-unitary) operator (Azizov [14]). Hint: Use the results of Exercise 16 on §1, Exercise 6 and 2.Formula (6.29). 13

Let A E St(./P; ko) and Im ko < 0. Prove that A admits extensions A =Ac E St(.M; Xo) with exit into some J-space .W D .W'. Moreover the following conditions are equivlent: a) A admits x-regular ko-standard extensions A = A `;

b) A admits regular ko-standard extensions 4= c) A admits maximal J-dissipative extensions A E St(. Y; ko) with (Azizov [14]).

Hint: Use Theorem 2.18 and the result of Exercise 6.

Xo E p (A )

266 14

5 Theory of Extensions of Isometric and Symmetric Operators Let .w be a J-space, N' +O 1P a J-space, where J = J Q+ 1, and let A be a J-symmetric operator. Prove that A admits Jself-adjoint extensions (cf. Exercise 13).

Hint: Use the scheme A -* JA -. JA O+ - JA -* JA O+ - JA = J JA 15

- JA.

VV= let the (.YYt , .W1 )-decomposition and Let V E St(.YPI, IY2 ), Jo [+] J, [+] V2 and the (.Yr z , .Wz )-decomposition W p = .moo [+] .-?, [+],R2 have the

property V11; = , ,, i = 1, 2. Then the following conditions are equivalent:

a) the operator A has at least one (J,, J2)-unitary extension; b) there is a (J,, J2)-unitary operator W such that WJv= Rv and WJ, = -R,;

c) there is a (J,, J2)-unitary operator U such that UJv=.wand UJ2=.-R2; d) there is a (J,, J2)-isometric operator V, mapping 91F1i on to .1,1.11 and in the Krein space VJ2 [ + ] V, (J i1i )2 there is a dual pair (Y" Y'-) maximal in that space and such that SP± fl VJz= (B) and Y'± fl V,(Ji1))2= (0) (Azizov [14]).

§3

Generalized resolvents of symmetric operators

Before describing generalized resolvents of J-symmetric operators we introduce and prove a number of auxiliary propositions which are, incidentally, of some independent interest. One of the concepts generalizing the 1

concept of an `extension' of operators is that of a `dilatation' of operators. An extension is a process which can take place both within the limits of the given spaces or with emergence from them; but a dilatation takes place necessarily with emergence from the original spaces. Definition 3.1: Let YW; (i = 1, 2, 3) be Hilbert spaces, and let T: ,-'de2 be a bounded operator with JT= .YW1. We call a bounded operator T:.Yy1 O+

O+

Y3 a dilatation of the operator T if T= 11

T;;jj i;= 1,

where T,1 = T and T21, T12, T22 are operators such that T12 T 2 t T21 = 0 when

n = 0, 1, 2, .... If also the Wi are J;-spaces, and if T is a (J1, J2)-unitary operator with it = J, O J3, JJ = J2 O J3, then we call T a (J,, J2)-unitary dilatation of the operator T We say that such a dilatation is x-regular if .W3 = II,Y with x negative squares; we shall call 0-regular dilatations regular.

Remark 3.2:

If under the conditions of Definition 3.1 Tis an extension of the

operator T (this is equivalent to the equality T21 = 0), then, clearly, T is a dilatation of the operator T. However, not every dilatation is an extension. We shall convince ourselves of this later, for example, in Theorem 3.4. Lemma 3.3:

Let T:.-WI -' X2 be a bounded operator with JT =,W,, and

be a dilatation of . let T: -W, 2 1Y3 3 T: .Ye,_O .W'3 O+ J 4 , Y'2 ( W3 O+ tea be a dilatation of

it,

and

let

the operator T.

Then T is a dilation of the operator T By hypothesis the operators T and T admit the follwing matrix represen-

§3 Generalized resolvents of symmetric operators

267

tations relative to the corresponding decompositions: T= II T%JII?J=I,

T=IITJII'J

with

T21=0,

and I

I T3 I I ?= I T33II T 3 j 1

1

=o = 0 (n = 0, 1, 2, ...).

Hence II Tu IIJ=2(II T,J 11 J=2)"II TJ1 IIJ=2 n-I

T12 T22 T21 + T13 T33 131 + T13 k=1

T33T32Tzz ` kT21

= 0 when n =0, 1, 2, ..., and it only remains to use Definition 3.1.

Theorem 3.4: Let T:,W, .W'2 be a bounded operator acting from a J,-space WI into a J2-space i 2 with CA T ='I. Then there is a J3-space W3 and a (J1 (@ J3, J2 O+ J3)-unitary operator T:.O, O+ W3 - -W2 O+ .03 which

is a dilatation of the operator T Let J, - T*J2 T = Jo I J, - T*J2 T I be the polar decomposition of the selfadjoint operator J, - T*JzT, let moo =3j, - T'J2T and let Jo = Jo I .Wo

Then Jo = Jo = Jo 1. We bring into consideration the Jo-space WO' = Q+

0

-.W o', where ,moo

= Yeo and Jo= (

k=0

JAkt, Jak)= Jo (k=0, 1,2,...),

and we define an operator T, :, O+ 06' -'2 O+ 0o by the formula

T1(x;xo,xl,...)=(Tx;IJI-T*J2TI1i2x,xo,X1,...).

(3.1)

By construction T, is a continuous operator with IT, = 'Y, 0,06' and it is (J1 O+ Jo, J2 O+ Jo)-isometric, i.e., T, is a (J, Q+ Jo, J2 O+ Jo)-semi-unitary

operator. The subspace moo is invariant relative to Ti, and therefore T, is a dilatation of the operator T. By virtue of Theorem 2.18 there is a J-space .W

such that the operator T, admits a (J, O J3, J2 O J3 )-unitary extension T: .01 O+ .3 -'2 O 03, where .3 = Wo O+ ' and J3 = JO' O+ J. It only remains to use Lemma 3.3. Corollary 3.5: For an operator T under the conditions of Theorem 3.4 to admit a regular dilatation it is necessary and sufficient that it be a (J,, J2)-binon-expansive operator. El The necessity follows from Exercise 34 on 2.§4. The proof of sufficiency can be carried out using the same scheme as in the proof of Theorem 3.4 taking into account that Jo = I I iJ, - T'J.T and that the operator T, is

268

5 Theory of Extensions of Isometric and Symmetric Operators

(J, O+ J6, J2 O+ J6)-bi-non-expansive. Consequently, by virtue of Theorem 2.18 we can put J = I. 2 Here we shall generalize the concept of a dilatation (see Exercise 4 below) and investigate special classes of holomorphic operator-functions.

Let T(µ) be a function holomorphic in the neighbourhood of 0 with values in a set of continuous operators acting from a space .I', into a Definition 3.6:

(B space .W'2 with VT(,) _ . 1, and let T= II T,, II ?i=1 :. 3 -,02 (9 '3 be a bounded operator with 1'r = 1Y1 O+ J °3. We shall say that the function T(u) 1

is generated by the operator T (or that the operator T generates the function T(µ)) if in some neighbourhood of zero T(µ) = T + µT,2(13 - µT22)- `T21.

Definition 3.7:

In Definition 3.6 suppose the .Yt'i are J,-spaces (i = 1, 2, 3) and

that T is a (J, O+ J3, J2 O+ J3 )-unitary operator. Then we shall say that the

function T(µ) belongs to the class H(MI, .2). If, in addition, .W'3 is a Pontryagin space with x negative squares, then we shall say that T(µ) belongs

to the class fl W',,.Y'2). (In cases where it will not cause confusion, the symbols 01, W2 will be omitted from the designations of classes). Lemma 3.8:

Let T= I I Tii

11 i=

i :.Yf, 0 JP3 - W2 O+ .k3

and

J4

W1

T=IITUII1i=i:.Y,

be bounded operators with #T = .N, O+ .YY3, VT= .-W, O+ .Yr'3 O+ .Yea, and let

T be a dilatation of the operator T Then T and T generate the same operator function T(µ) : Ye,

Ye2.

In accordance with Definition 3.6 we have to verify that T11 + µTlz(I3 - µT22) - `Tzl

= T + It II Tii II i=z(I3 G4 - µ II Tii

II'i=2)

`II

Ti,

II'=2.

(3.2)

The condition 'T is a dilatation of T' is equivalent in the present case to the system of equalities T3 T 33 T3j = 0 (i, j = 2, 3), and therefore II T,iHI]=2(II TiiII'i=2)'II Ti li=z= T12Tz2T21

(k = 0, 1,2).

It only remains to note that in a sufficiently small neighbourhood of zero the equality (3.2) is equivalent to the coincidence of the series k= =0

µkT12T22T2,

and

k=0

µk11 TUIIJ=2(11

TiiII'i.i=2)kII Ti, II' =2.

§3 Generalized resolvents of symmetric operators Lemma 3.9:

269

Let T(µ) :.WP, --+ k2 be an operator function holomorphic in

the neighbourhood of zero. Then there is a space IY3 and an operator T= II T, II ?i=1 : h"1 T

O+

3 such that T(µ) generates the operator

--W2 O+

3

We consider the function, holomorphic in the neighbourhood of zero,

K(µ) = (T(µ) - T(0))µ with µ 5;6 0 and K(O) = T'(0). By Cauchy's formula we have for it, when r > 0 is sufficiently small, K(µ)

tai

r-µ

t1=

2ir

N)

A

K(rfl d(' = 1

K(reI

d4p

(I it

I < r).

µ

J

Let .03 be the Hilbert space of weakly measurable functions x3 (e "0) on [ - a, n] with values in .Y2 and with the square-summable norm IIx3(e'")I12dtip
let T22 be the operator of multiplication in .1Y3 by r-'e-"°, let

T21x, = K(re' )xi when x, E k,, let A

T12x3 = 1

x3 (e' J

d4p when x3 E --W3, and let T = T(O).

Then the operator T = II T, II ?i= I defined on Wi O+ .l'3 is bounded, acts into ./r'2

.IY3, and T(µ)= T(0)+µK(µ)= Tu+AT12(13-µT22)-'T21.

We summarize the results set out. Theorem 3.10: If T(µ) :.)Y, .Y(2 is an operator function holomorphic in the neighbourhood of zero, then T(µ) E H. 11

It suffices to compare Lemma 3.9, Theorem 3.4, and Lemma 3.8.

In this paragraph we introduce a number of results and constructions on the basis of which we shall obtain a criterion for T(µ) to belong to the class II'.

3

Definition 3.11: A function K(X, µ) of two variables, defined on A x A, where A = A* C C, and with values in a set of continuous operators acting in a Hilbert space .# and defined on it, is called a Hermitian kernel if

K(X, µ) = K*(µ, X); a Hermitian kernel K(X, µ) is said to have x negative squares if for arbitrary finite sets (X;); C A, ((;i12 C C, and (x;); C ./P the quadratic form t.l=,

(K(X;, X )x;, xi)Zi i

(3.3)

270

5 Theory of Extensions of Isometric and Symmetric Operators

has not more than x negative squares, and for at least one set has exactly x negative squares. In particular, if x = 0, then K(X, µ) is called a non-negative Hermitian kernel. El Let ,' be a J-space, let F(k) :.0 W' be a function holomorphic in the neighbourhood WF of zero with values in a set of continuous operators with /JF(X) _ ,Y, and let K(X, µ = J[F(X) + F`(a)]/(1 - Xµ) be a Hermitian kernel

having x negative squares. We shall set up by means of the kernel a correspondence between the function F(X) and a certain Pontryagin space II, (F), we introduce a continuous operator P :.° II, (F), and a certain 7r-semi-unitary operator in and we express the function F(X) by means of them. Let X E v(F, let ex be a symbol, and let x>, be a vector in .,Y. We form a linear set S' consisting of formal sums f = Ex ( Y/, eXx>, in which only a finite number

of the xa i;e 6. For elements f and g = EX E

e),g), we define a form

Y/,

If, glo = Ea.µE 111, (K(X, µ)x>., yµ). By hypothesis this form has x negative squares, and if P is degenerate and Yo is it isotropic part, then the completion of the factor-lineal Y/Yo will be a Pontryagin space with x negative squares

(cf. Exercise 6 on 1.§9)-and we denote it by fl,(F). From now on we shall identify elements from 9 with the corresponding elements from £lYo. On the set

1v= j.f If= aE Yj14 e),xxEY, Y xx=9} XE 14

we define an operator V : Vf= EXE 111, XE)IXX- Since

[Vf, V.f]o= >

x

X.µ E Y/,

X.1+ E Y/,

1-Xµ =

xµ

x

p E Y/

X E Y/, L

F(X)x,\,

F(X) + F`(µ)lxx, 1- Xµ J

xµ

J

([F(k)xa,x,,]+ [xa, F(µ)xµ])

[F(X)x),, X E Y/,

\\1 J

1

µ E Y/,

J

µ E Y/, 1- XtL. xµJ

['Z/,

+

kµ

1-Xµ 1

µ E Y/,

I

xa, F(y)x, ]

XX, F(A)x,

X E W,

=[f,flo, Visa 7r-isometric operator in II, (F). Its range of values fit v consists of vectors g = EX E Y/, e,,ya E 9' such that yo = 0 and Eo , a E Y,, y,\/ X = 0. This set is dense in

n, (F). For, if e; x -' eox, ('e),x

--+ 0 and x E ., then, using Exercise 13 on 1. §9, we obtain 0, and therefore any element E,, E Yn exx7, E ' (xo = 0) is

approximated by the elements EXE Y/, eaxa - (et Eo4 XE Y,, x>/X E R v, and since

Y' = II, (F), it follows that ,;? v = rI, (F) also. It follows from the last relation that is non-degenerate, and this implies that V is invertible and hence

(taking 2.Corollary 4.8 into account) that the operators V and V-1 are

§3 Generalized resolvents of symmetric operators

271

continuous. We keep the same notation for the closures of these operators, the second of which will be 7r-semi-unitary. We bring into consideration an operator IF : Ye -p 2' by putting Fx = (1 /, 2)eox, x E M. For any x, y E Y and X E WF we obtain [Fx, Fx]o = [(F(0))Rx, y], [rx, exy]o = (1/,2) [F(0) + F`(X))x, y], and therefore IF is a continuous operator (see Exercise 6 on 1 §9). Consequently there is the operator r': rl (F) .Y and

FT = (F(O))R,

r `c),x = (1 ,,!2) (F` (0) + F(X))x.

(3.4)

Since (see 2.Definition 6.6) the spectrum of the ir-semi-unitary operator V ', with the exception of not more than x eigenvalues, is situated in the disc (t t 15 1), the resolvent (V - XI) - ' exists everywhere in the disc (X I I X I < 1) with the exception, possibly, of not more than x points. Since

(V- XI)(e),x- cox) = Xcox, we have r`cxx-r`eox= XT`(V- XI)- Icox, and so we obtain from (3.4)

F(X) = i(F(0))r+ F`(V+ XI)(V- VI)-'F.

(3.5)

We sum up the foregoing argument:

Let V' be a J-space, let F(X) be an operator function holomorphic in the neighbourhood of zero and taking values in the set of continuous operators defined on ', and let the kernel K(X, µ) = J(F(X) + F`(µ)) (1 - X) have x negative squares. Then: Theorem 3.12:

1) F(X) extends, preserving these properties, on to the disc [X I I X I < 1) with the possible exception of x points; 2) there is a space II,,, a bounded operator r : -+ II,,, and a a-semi-

unitary operator V ' : H - H. such that the equality (3.5) holds for all

X a(V),IXI<1. E Remark 3.13: In Theorem 3.12 the operator V can be regarded as ir-unitary since, by virtue of Theorem 2.18, it admits regular extensions V and Mr=(V-XI)-'I -f r. (V-XI)-`I The proof of the main Theorem 3.16

in this section depends on the following two lemmas, the first of which is a direct consequence of

Theorem 3.12.

-+ r be an operator function holomorphic in the neighbourhood of zero, let the kernel L(X,µ)= J - T*(µ)JT(X)/(1 - Xµ) Corollary 3.14: Let T(X) :

have x negative squares, and let 1 E p(T(O)). Then T(X) E II k. Since 1 E p(T(0)), there is a neighbourhood -V/o of zero such that 1 E p(T(X)) when X E ?lo. We put F(X) = (rzI + a T(X )) (I - TX)-' when X E 40/0. The function F(X) is holomorphic on Rio, and since T*(µ) JT(X) F(X) + F`(µ) _ (I - T(1\)) , 1 - µX - 2 Re a(1- T *(µ)) ' J- 1 1

272

5 Theory of Extensions of Isometric and Symmetric Operators

so when Re a > 0 the function F(X) satisfies the conditions of Theorem 3.12. Let II,,, r and V be the same as in Theorem 3.12, and suppose (see Remark 3.13) that V is a IIx-unitary operator. Since T(X) is expressed in terms of F(X) by the formula

T(X)_(F(X)-czl)(F(X)=al)-' = 1-2 Re a(F(X)+al)-', so, using (3.5) we obtain

T(X) = (F(0) - al) (F(0) + al) + X(24 Re a (F(0) + al)-'P ` V- (1- X (V-' - 2I' (F(0) + al) -'P ` V-'))-' x (2,jRe a F (F(0)) + aI)-1).

We define in the space If O+ 11, an operator T= II Tai II ?i= 1, where

T = T(0)=(F(0)-dI)(F(0)+al)

1,

T2, = 2, Re a (F(0) + aI) -',

T22 = V- 1 - 2I' (F(0) + al) - 1I' ` V-'.

T12=2,Re a(F(0)+a!)-'I"V-',

This operator generates the function T(X), and it is verified immediately (takinn the equality FT = (F(0))R into account) that T`T = T`T = I, i.e., T is a (J + operator. Let To: . , -.-*12 be a continuous operator, with 9'T, =,W,. There will be a x-dimensional space II_ and a (J(, J2)-bi-non-expansive

Lemma 3.15:

operator T= 11 Ti; Hjz j=, acting from the J;-space i = . , O+ II_ into the J2'-space W2=.W2 O+ II_, with J; = J; O+ - I_ (i = 1, 2), and such that T11 = To, if and only if the operators J, - To J2 To and J2 - To J, T* have the

same number x, 5 x of negative eigenvalues (taking multiplicity into account). Under these conditions the operator To is expressible either in the form To= RW, where W is a (J,, J2)-bi-non-expansive (J,, J2)-semi-unitary

operator, and R :.i'2 - W2 is a bounded operator with Ker R J Ker W` and the set [C\(IR U iIR)] fl a(R) is no more than countable, in the form To = W' R', where W' is an operator (J2, J, )-conjugate to a (J2, J,)-bi-non-

expansive (J2, J, )-semi-unitary operator, and R' :.1 -., is a bounded operator with Ker R" D Ker W' and the set a(R) E [C\(1R U iR)] is no more than countable. El Suppose the subspace II_ and the operator T indicated in the Lemma exist. In accordance with 4.Theorem 1.12 the operator T` admits a (J2, J, )polar decomposition T` = V`S, where without loss of generality (otherwise we would consider the operator To instead of To) we can suppose that V is a

(J,, J2 )-semi-unitary (J,, J2)-bi-non-expansive operator, and Sc has the properties of a J2-module indicated in 4.Theorem 1.12. Hence T = S V. Let P;

be an orthoprojector from .'I on to Y;, i = 1, 2. Then To = P2 S VP1 I .%P, .

§3 Generalized resolvents of symmetric operators

273

If

H-

S = II S.! II ?J= 1 :,-W2

--W'2

n-

and

fI_'.V2 1-1V= are the matrix representations of the operators S and V, then To = S11 V11 + S12 V21. Just as in the proof of the implication b) - c) in Theorem 2.18, we prove that the operator V11 I Ker V21 admits a (J1, J2 )-semi-unitary (J1, J2)-bi-non-expansive extension W, and we let P be a J2-orthoprojector from W2 on to Rw. Then VaiII?J=1:.-W1

TO= (S11 P + S11(V11 W` - P) + S12 V21 W`) W = R W. By contruction W has

the properties indicated in the Lemma, and Ker R 3 Ker W`(= RV). We verify that a(R) fl [C\(IR U iIR)] consists of not more than a countable set of points. Since the operators S11(V11 W` - P) and S12 V21 Ware finitedimensional, it is sufficient by virtue of 2.Theorem 2.11 to establish that the set

a(S11P) fl [C\(IR U iIR)] has the property mentioned. In accordance with Corollary 3.5 the J2-selfadjoint operator S, which is J2-bi-non-expansive, admits a regular dilatation S which is a J2"-unitary operator acting in the JtZ = W2 O+ II- O+ fl,, where J2" = J2 O+ - I_ O+ I+. Let Yez = 3 w [+] RVI, and let 9= II S;; II i;= 1 be the matrix representation of the operator S relative to this decomposition of 2 . Since a(S11P) - (0} = a(PS11 P) - (0), and PSP I Mw = S11 (= 911), it is sufficient to verify that the set fl [C\(IR U iR)] is no more that countable. Since the operator S is J2-unitary, it follows that 11 - 921, = 92,,§21. Let Xo be the boundary point of the spectrum of the operator I1 - 92, 1. Then there is a sequence (x,, } E Mw such J2" -space

that 0
S21Szl which is a (I- P)* Jz (I - P)-selfadjoint operator. Since W is a (J1, J2)-bi-non-expansive operator, the J2"-orthogonal complement Iw, to .l?w is a Pontryagin space with x negative squares. Therefore (see 2.Corollary

3.15) a(I, - S21)fl (C\IR) consists of not more than 2x eigenvalues (taking multiplicity into account), and therefore fl [C\ (R U iIR)] consists of a finite number of eigenvalues of finite multiplicity. Thus To = R W1 where R and W have the properties indicated in the Lemma.

Since T is a (J;, J2)-bi-non-expansive operator, there is, by virtue of Corollary 3.5, a Hilbert space fI+ and a (J,, J2)-unitary operator T= II T;; II ? i= 1

acting from the Jspace '1 = .,'1 (1 (II_ O+ fI+) into the

J2-space ie2 =..Y2 O+ (H_ ( f1+ ), where J; = J; O+ ir, and such that IF,, = To. The equalities J1 - To J2 To = T2, rr. T21

and

I_ + I+,

J2 - To J, To = T127r. T12

can be verified immediately. Consequently (see Exercise 14 on 1§9) the

5 Theory of Extensions of Isometric and Symmetric Operators

274

operators J, - 7o J2 To and J2 - To J, To* have respectively x 15 x and x2 < x negative eigenvalues (taking multiplicity into account). We recall that x, and x2 coincide with the number of negative squares of the forms [(I, - ToTo) , ], and [(I2 - To To) , ]2 respectively. Since

I, - To To = II - W`R`RW = W`(I2 - R`R)W, and Ker R 3 Ker W`, and Ker W` is a uniformly positive subspace, it follows that x, coincides with the number of negative squares of the form [(12 - R`R) , ]2. We note also that

from the condition Ker R 3 Ker W` the equality 12 - To T` = I2 - RR` follows.

Let X be a linear operator; we shall denote by the symbol X, the operator e' '`X (SO E FR). It is easy to see that

I2-R,(R,,)`=I2-RR`.

and

Since the set o(R) fl [C\(IR U iIR)] is no more than countable, there is a number po such that 1 E p (R,,). Consequently I2 - R `R = (12 - R,,)` [ (I2 - R,0) ' + (12 - (R,0)`) -' - 12l (12 - R,0),

12-RR`=(12-RPo)[(12-R,,,,)-'+(12-(R,p,)`)-'-12](12-(R,)`), and therefore

12 - R`R = (12 - (Rp)`)(12 - Rw0))-'(I2 - RR`)(I2 - (R,0)`)-'(I2 - R,,,), which implies that the numbers x, and x2 are indeed the same. Conversely, let the operator To be such that each of the operators J, - T, J2 To and J2 - To J, To* has exactly x 1 5 x negative eigenvectors (taking

multiplicity into account). We shall first suppose that dim J", = dim W2. We bring into consideration the operators TV) = To Fn,

where F" =

+1

P1 + rn

P1

(n = 1, 2, ...)

For sufficiently large n the negative spectrum of each of the operators J, - To)*J2 To") and J2 - T(n)J,To") consists of exactly x, (taking multiplicity into account) eigenvalues and these operators are boundedly invertible.

We denote by J;") and j2(n) respectively the unitary parts of the polar decompositions of these operators, we put Pi,, = z (I; J;")), i = 1, 2, and we ± bring into consideration the j;')-spaces it- ,") = .YP; O+ .Ye,, J;") = J; ( j(n) and the operators T(n)

= II

7{n) II2 . . Wi(n) t.J= l .. ,j

Wi(n) 2

where ,f

71

Tin) = Ton)

-I J, -

72(n) = 1

T2(121)

1/2 yn, n) = I J2 - l on) J17pn)*I

7n)* J2 T(n) 1/2, o

o

I 0 J(111)1 JI - P011)*

0

0

J2-

0

0

where Vn is a unitary operator mapping Ye, and to .02 (its existence follows

§3 Generalized resolvents of symmetric operators

275

from the fact that dim .)Pi = dim .IY2) and moreover mapping P,;,,. 1 on to P2.,,.Y('2. It can be verified immediately that T(' is a (J,("), J2("))-unitary operator. Consequently (see Exercise 34 on 2.§4) the operator f,(n)

= I Ti") I?i=1:.%t'1 +

P 1,n..w'1,

where

Tii)= T(i), Tzi)= P1,nTii), Tiz = Tiz)Pl,n, and f = P1,nTzi) P1, n, - Ii,n, J2 O+ - Ii,n)-bi-non-expansive operator. The operators P;-" converge in norm to P;,o when n -' oo, where P;;o = (I - P,,o - J;, o) (i = 1, 2), is a (Jl

i P1,o is the orthoprojector on to Ker(J1 - T'J2To), P2,o is the orthoprojector on to Ker(J2 - To J1 To), and Jl,o and J2,o are the partially isometric parts of the polar decompositions of the operators J1 - To J2 To and J - To J1 To* respectively. We choose Vn so that the operators VnPi,n converge in norm to VPI;o, where V: Plo 1 -' Pzo.W2 is an isometry. Then the operator Pj,oa)l

T= II TjiII?.i=:,Wl(@ with T= To, T21 = (Pl.o(T0J2T0 -

T12 = (P2.o(To J1 To - .12)P2,o)1/2 VP1;o, J1)P170)1/2'

T22

= T 'P1.oToJ2T12

will indeed be the required (Ji, J2)-bi-non-expansive operator

(J;= J; G) -Ii,o,i=1,2). We now suppose that dim .W1 ;e dim a2, and that dimensional Hilbert space with dim ,4

,

'

is an infinite-

max (dim .1, dim ,Yt'2 l . We consider

an operator TV acting from the (Jl O+ I)-space k1 O+ ' into the (J2 O+ I)space .2 O+ . and which is an extension of the operator T, and moveover TV I .W = I. The operator T1S satisfies the same conditions as the operator To,

and dim(.1 O+ .) = dim(,Y2 0 .0), and therefore the proposition proved above holds for it. It only remains to use the result of Exercise 34 on 2.§4.

Theorem 3.16: Let T(X) = Z' O X' T1 bean operator function holomorphic in the neighbourhood of zero. Then the following assertions are equivalent:

a) T(X) E rl';

b) the negative parts of the spectra of each of the operators J;n) - T(n)* JZn)T(n) consist of not more than x negative eigenvalues, and the

negative parts of the spectra of the operators Jf O) - T (O)* J1°) T(O) and J)°) - T (O) J1 T (O)* consist of an equal number x 1 <, x of negative eigenvalues

(taking multiplicity into account), where J°r = Jk, Jkn) = Zk=o O+ Jk, and T'n) = II T1 II"i=o are Toeplitz matrices acting from the J(n)-space ,y,n) into

5 Theory of Extensions of Isometric and Symmetric Operators

276

the J2(")-space .YPZand Tij = 0 when i > j, T;; = Tj_ 1 when i <, j, k(n)_ yek

.%Pko)_yek,

L

...

i,j=0,1,2,...,n;

(DWkg J

n+l

c) the kernel [J1 - T*(µ)J2T(X)]/(1 -µX) has not more than x negative squares, and the negative parts of the spectra of each of the operators J1 - To J2 To and J2 - To J1 To* consist of x, < x2 eigenvalues (taking multiplicity into account).

a) - b). Let T(X) E rIx, i.e., there is a Pontryagin rx-space rrx = rI_ O+ rI+ with dim rI_ = x < oo and a (J1 O+ rx, J2 O+ rx)-unitary O + Hx - M22 operator T= I I T;jjj 1 j= 1 : rIx such that in the neighbour(

1

hood of zero T(X = T11 + X T12 (Ix - X T22) _' T21. Hence

To =T11,T;=T12Tzz1T21) i=1,2,..., and

Jln)- T(n)*JZn)T(n)= 7'(n)*J(n)T(n), where T<") = II

Tij II ;'j=0: Jyj 0 ...

o

,

rrx

o ... o

n+1

rrx

n+1

is a diagonal matrix with T, = T22T21 (i = 0, 1, 2, ...), and J(") = I JU

0:1Ix

...

rIx

n+1

rrx O+ ... O+ rIx, Jij = rx, n+1

i, j=0, 1,2,..., n; n=0, 1,2,.... Since the selfadjoint operators J(") have not more than x negative eigenvalues, it follows from Exercise 14 on 1.§9 that the operators J;") - T(")* J,(")7{") also have not more than x negative eigenvalues (n = 0, 1, 2,. ..).

The validity of the assertion that the operators

Ji0)

- T(0)* J2 T(0) and

J2(0) - T(0)J;0)T(0)* have the same number of negative eigenvalues follows from Lemma 3.15 and Exercise 34 on 2.§4. b) - c). Since o( J;") - T(")* Jz")T(")) fl (- oo, 0) consists of not more than x negative eigenvalues (taking multiplicity into account), and this is equivalent to the fact that the forms ((.Iin) - T(n)* JZn)T(n))X(n), x("))

have in, O+ ...

O+ ,Ye'

+rl

not more than x negative squares, it follows that for arbitrary finite sets of vectors (x1} " C .,Y1 and arbitrary complex numbers (Xj] 1 the form

((Jin) - T(n) J2nT(n))x,(.s), x,j'4), r, s, p, q

§3 Generalized resolvents of symmetric operators

277

where

O

X; s) = (Xr, X5Xr, ..., X s Xr) E .,Y1

... O

.

1,

(n + I) times

s= 1,2,...,t; r= 1,2,...,m, has no more than x negative squares. This form can be rewritten as

((1_X)(JiXrXP)i

1

r.s.P,q

1 - X5X

+Z

(1 - Xs

J2T x , Xr)1

i.J=o

((1 -

V= qTJ*) J2 \i

r,,p,q

0

xsT,/) Xr, XP)

1

1 - XSXq XP)1 1\

I - X, 1q

r, s, p, q

n-ljn-1

n

s

+

i.J=0

^9 (_1 - XsXq

(X lT!)J2(X'sTi)Xr, XP

To prove our assertion it suffices to verify that the second sum over r, s, p, q tends to zero as n oo if we take the (Xi) i in a sufficiently small neighbour-

hood of zero. Without loss of generality we shall suppose that

II Xr II = 1

(r=1,2,...,m),andIX <e(s=1,2,...,t)ande'12II TTII <, 1(j=1,2,...) (the latter is possible because T(X) is holomorphic in the neighbourhood of zero). Since there is only a finite number of terms in r, s, p, q, we shall verify that each of them tends to zero as n oo; indeed, n

X'nXn(J1Xr, XP)1 + Z (Xs

Ti)Xr, XP)1

i,j = o I - XsXq

2n+

S

n

Z

£2n+(j-3i)/2

i''=o

0

1 - c2

as n - oo.

c) - a). In accordance with Lemma 3.15 we can suppose without loss of generality that To = R W, where R and W have the properties indicated in that lemma. Let tp E [0, 2a) be a number such that 1 E p(R,,). We consider the function which is holomorphic in the neighbourhood of zero. This function satisfies the conditions of Lemma 3.14 and therefore R,(X) E II"(,'2), i.e., there is a aX-space II,, and a (J2 O+ ak)-unitary operator R = II Rii. vll ?i= l :.f(2 G IIX --

ED n.

such that in the neighbourhood of zero R,(X) = R11,,, + XR12,,,(I, - >,R22.,)-IR21,,.

5 Theory of Extensions of Isometric and Symmetric Operators

278

Hence,

T(X) = R,(X)W-,= R11.,W-.c+ XR12,,,(I"- XR22,,)-'R21,,W and therefore T(X) is generated by the matrix T' = II T11II i= 1:.W1 on"-..YP2 orlk, where

Tit = R11,,. W-., Tie= R12,,, T21 = R21, W-,, T22= R22,,,. By construction T' is a (J1 O+ r", J2 O+ r" )-semi-unitary (J1 O+ r", J2 O+ r" )-

bi-non-expansive operator. By virtue of Corollary 3.5 it admits regular dilatations, and it follows from Lemma 3.8 that each of these dilatations generates the function T(X), and therefore T(X) E II". Now let W 1 = W2 = . be a J-space, let T(µ) :.W -p ,W be an operatorfunction holomorphic in the neighbourhood of zero, let 4'o be a Jo-space, and let T= II T, II ?r=1 : ,Y e ,Yo - W 0 o be a J-unitary operator generating the function T(µ), where J= J+ Jo. Definition 3.17: The J-unitary operator T= 11 T1Ij i1= 1 will be called simple if the operators T and T22 have no common completely invariant subspaces.

Theorem 3.18: Every operator function T(#): -.71' holomorphic at zero is generated by a certain simple J-unitary operator T = 11 T.i II?i=1 :.(0

-.R 0 .7(o.

Moreover, if T(µ) E IT, then a Pontryagin space with not more than x negative squares can be taken as ./1'o.

Let .73 be the J3-space which exists by virtue of Lemma 3.9, and let T' = II T,i 11 1 i=1 : -W 0 -Y3 - . W 0 3 be the (J (@ J3)-unitary operator generating the function T(µ) (if T(µ) E IT, then we take as '3 a Pontryagin space with not more than x negative squares). The subspace .1P" _ C Lin ( T' "01

is completely invariant relative to the operator T', it contains

./P, and the operator T' is a dilatation of the operator T" = T' I .7(", and therefore (see Lemma 3.8) the operator T" generates the same function T(µ), and moreover T" is an isometric operator in the indefinite metric induced by

the (J (@ J3)-metric in 7". Let P be the orthoprojector from .0" on to its isotropic part. Then the space ,Yo' = (I - P).,Y" contains . and is a nondegenerate (J O+ G)-space :.(o = 0 W', and the operator T" is a (J (@ G)unitary dilatation of the operator To = (I - P) T" I.7(6 and therefore, again by virtue of Lemma 3.8, To generates the same function T(µ). As the space o we take the completion of ,Y' relative to the norm (I G I x, x)1 /2 (cf.. I .Proposition

6.14); as Jo we take the extension of the isometric part of the polar decomposition of the operator G on to .-Wo; and as T= 11 T, Il zj=1 we take the extension by continuity of To on to .7 0 YW o. It is clear that the operator T

§3 Generalized resolvents of symmetric operators

279

generates the function T(µ) and is f-unitary, where J= J OB Jo, and o = C Lin ( T".-*'] We now verify that T is simple. Let . be %r O+ the common completely invariant subspace of the operators T and T22. Then Y'I1]

is completely invariant relative to T and .X C q'111, and therefore

C Lin(T"k)°°_ C Y I1], i.e., 91111 =.Ye O+ .Yo and therefore £_ (0). We note that if .W'3 is a Pontryagin space with not more than x negative squares, then °o also has this property.

Definition 3.19: An operator-function R;, holomorphic in a neighbourhood of the point k is called a generalized resolvent of a J-symmetric operator A acting in a J-space . if there is a J1-space .1 such that the J Hermitian operator A in the f-space O+ .,Y1 (where j = J (D J1) admits a fselfadjoint extension A with ko E p(A) and R1, = P(, T- XI)-' I , where P is 4

='

the orthoprojector from ie- on to. We call such a function R1, a x-regular generalized resolvent if some Pontryagin space with x < oo negative squares can be taken as .,Y,, and a regular generalized resolvent if x = 0. Definition 3.20: We denote by the symbol II>, (A) the set of all operatorfunctions F(X) :.W' .W, holomorphic in a neighbourhood of the point Xo, which can be expressed in the form F(k) = F1 1

where F(X) = I

- xo F( _ kok ko k 12\I J1)-unitary

kok - k0

+kok -

F,i II ?i=1

\

Fzz l

Fz 1,

extension of the operator

is a (J O+

IA.x,, in the (J (@ J1)-space f O+ . 1; .01 is a certain J1-space; if .1P1 is a Pontryagin space with x negative squares, then we denote this set by II

'A _

rIL(A)

It follows immediately from the definition of the set 11x.(A) that for every k from a certain neighbourhood of the point X0 the function F(X) (EIIao(A )) is a J-bi-extension of the operator IA,XO and the operator-function

G(µ) = F(I )'°I z(1

\

ko - Xoµ

µ) I E H

l

is holomorphic in a neighbourhood of µ = 0. Conversely, suppose that for every k from a certain neighbourhood of the point Xo the operator-function F(X), holomorphic in this neighbourhood, is a J-bi-extension of the operator IA,a,,. Then it belongs to the set II1,,,(A ). For, in accordance with Theorem 3.10 G(a) E H and therefore kok - k0

F(k)=F11+kok-

.0

F

1z

- x0 Fzz 1 I I1 _ ko kkok-xoJ

1

Fz1 ,

and moreover by virtue of Theorem 3.18 we can regard the (J O+ J1)-unitary operator F= II F, II Z;=1 :.k' O+ .k1 -* .Y( O+ .,W1 as simple. (We note that we can take H, as.W1 if and only if G(µ) E H".) We verify that Fis an extension of

5 Theory of Extensions of Isometric and Symmetric Operators

280

the operatorlA,a or, what is the same thing, that F21x = 0 when x E WA - i. Suppose this is not so. Then there is a vector xo E MA - j,r such that F21x0 0, and therefore Y' = C LinI FX"J °m n w'1 * (0) and it is a completely invariant subspace relative to F and F22-and we have obtained a contradiction of the fact that the operator F is simple. Thus we have proved

Let F(X) be a function holomorphic in a neighbourhood of the point X0. Then F(X) E IIx, (A) if and only if, for every X in a certain

Theorem 3.21:

neighbourhood of the point Xo, F(X) is a J-bi-extension of the operator IA,X,. Moreover the conditions (F(X) E Hx'o (A )) and j G(µ) = FI' X01 z(1

are equivalent. The operator F be chosen to be simple.

µ))

E

n'}

F;; i;=1 generating the function F(X) can

Let a J-symmetric operator A E St(,W; X0) with Im Xo < 0, let Ao E St(, ,Y; X0) be a J-bi-extension of the operator A such that Xo E p(Ao) (on the existence of such an Ao see Exercise 10 on §2), and let Uo = K1,,,(A). 5

Theorem 3.22:

The relations R>, = (Ta - XI)-' and Ta=(X0-Xo)[F(X)Uo(UoUo)-i-I]-1+X I

describe all the generalized resolvents of the J-symmetric operator A in the neighbourhood of the point Xo when F(X) traverses the set In the neighbourhood of the point Xo we have R"

= El

[(Tµ-µ1)-1l`.

Let A be a J-selfadjoint extension of the operator A acting in the J-space w' O+ -Y,, and let Xo E p(A). It follows from Exercise 12 on §2 that (A - X01)-' =

1

- x0

[4.00(17C0170-1

- 1],

where U0 is a J bi-extension of the operator U = K,,o (A) coinciding on W' with U01 and Uo I, = (X0/4)11i and !>,, is the corresponding J-bi-extension of the operator IA,XO. In accordance with Hilbert's theorem for the resolvent R,,

1

-

P I - X_

Ilo (1 >,,,Uo(UoUo)-1 - I

Jof

[i U0(UoUO)-'I] I .i

Let Ik,,= II Fiijj?i=I be the matrix representation the operator operator U= has the matrix representation U= 11 U1ijj?i=1, where

Un = FnUo(UoUo)-', and Uiz>'o F,2, x0

.

Then the

i= 1,2.

§3 Generalized resolvents of symmetric operators

281

Therefore in a neighbourhood of the point Xo Hilbert's identity given above can he rewritten in the form r /F(X)Uo(UoUo) I F(X)Uo(UoUo) Xo -

where

Xo - xo

O

\

L

F(X) = F1 1

Xo X - X0

-1

Xo X - X0

+Xo X - Xo F12

IXo X - o

Fzz

Fzi

is an operator-function holomorphic in a neighbourhood of the point X0 which is, for every X from this neighbourhood, a J-bi-extension of the operator IA,X,, i.e., F(X) E II,,,,(A) by virtue of Theorem 3.21. In this same neighbourhood of

the point ko the operators F(X)Uo(UoUo)-' are J-bi-extensions of the 1 E o, (U) that where Rs=(T,-XI) ', Hence, I]-1 + X01. Tx _ (Xo - Xo)[F(X)Uo(UoUo)-' Conversely, let F(X)EIh,,,(A). Then F(X)Uo(UoUo)-` is an operatorfunction holomorphic in a neighbourhood of the point Xo which, by virtue of for all X from Theorem 3.21 is a J-bi-extension of the operator U= this neighbourhood. Therefore

operator U= K,,, (A ), and therefore it follows from

IEa,(F(X)Uo(UoUo)-')

/ T(µ) =

z

F( 1 X0 '

Uo(UoUo)-'

(l

X0-1,A))

is a function holomorphic in a neighbourhood of the point-,u = 0, and therefore '1 and a simple J-unitary operator there is a Ji-space

U=

U;;11 ;=, :

0

, --

.

0 .W'i, with J= J 0 Ji, such that U gener-

ates the function T(µ). It follows from the result given later in Exercise 11 that

we can take as U an extension of the operator U. Therefore by virtue of Exercise 16 on §1 U=I,,oUo(UoUo)-1, where Uo is a J-bi-extension of the operator U coinciding on r with Uo, and Uo I W', = Oo/Xo)h, and 1),0 is the corresponding J-bi-extension of the operator 1A.ao. Since U C Uand 1 E ac(U),

it follows, because the operator U is simple, that 1 E oc(U), i.e., there is a J-selfadjoint operator A such that U = Kx,,(A ). It follows from Exercise 15 on II§6 that A C A, and from the proof of the first theorem that

R,,=(T,,- XI)-', where

Ta=(Xo-&) T Q

I

+ XI

= (Xo- Xo)[F(X)Uo(UoUo)-' - I]-' + Xol. It

then follows from the definition of the generalized resolvent that

R, = (R,,)` = [(T;, - µl)-']` in a neighbourhood of the point Xo. Corollary 3.23: Under the conditions of Theorem 3.22 let A0 be a maximal generates J-dissipative J-symmetric operator. Then the function F(X) E a x-regular generalized resolvent, and conversely, every x -regular generalized resolvent is generated by some function from nx',,(A).

5 Theory of Extensions of Isometric and Symmetric Operators

282

Let the function F(X) E HZ,/(A ). Then

FIIX 12(1-µ))U0E H',

T

and the operator U appearing in the second part of the proof of Theorem 3.22

can be chosen to be a x-regular extension of an operator U= Therefore the generalized resolvent constructed there will be x-regular.

Conversely, let R), be a x-regular generalized resolvent generated by a x-regular extension of the operator A. Then U = Ka (A) = where U0 is a J-bi-non-expansive i-semi-unitary extension of the operator U coinciding with Uo on .W, and Uox = (Xo/Xo)x when xE I ,,. Consequently I>,0UoUo is a i-bi-non-expansive extension of the operator IA,ao and therefore, by virtue of Corollary 3.5 and Lemma 3.8, the function generated by this operator belongs to III (A ). Corollary 3.24: Under the conditions of Corollary 3.23 let J = I, x = 0. Then the function (TX - XI)-' admits a holomorphic extension from the neighbour-

hood of X0 on to C-. On C+ we put (Tx- AI) -' = [(TX- XI)-']`. This extension coincides in C+ U C- with the corresponding regular generalized resolvent of the operator A. By virtue of the result of Exercise 12 on §2

Ta=(X0-Xo)(F(X)Uo-I)-'+X01, and by virtue of Corollary 3.23 we can suppose that F(X) E i.e., there is a Hilbert .4" space and a unitary extension (@.;99, of the operator IAA, such that F= II FjiHI +i= I :,-W O+ X F(X)

= Fi +

.o

Xo

_Xo i - Xo

>-1\0 X - 1\o

F 1z I

X0

X - 1\0

\-i

Fzz

Fz i

Since

II Fzz II < and

),o

X - X0

Xo

X - .o

< 1,

it follows that F(X) admits a holomorphic extension from the neighbourhood of the point Xo on to C-. Moreover the extenson of the function z

G(µ) = F( X01 0 _ 01 E IV( 4 - Xoµ

< 1, and therefore II F(X)II < 1),

which, by virtue of 2.Remark 6.13, implies that the operators Ta are maximally dissipative for all X E C-. Therefore C- C p(TX) (see 2. Lemma 2.8),

and therefore (T,, - XI)-' admits holomorphic extension from the vicinity of

the point Xo on to C-. On C+ we put (T,- XI)-' = [(Tx- XI)-']`. On the other hand, the regular generalized resolvent R,, of the operator A

§3 Generalized resolvents of symmetric operators

283

defined on C U C- does coincide in a neighbourhood of the point X0 with the

function (T,\-XI)-', and therefore R a = (T, - XI)-' when X E C- and µEC+.

Exercises and problems 1

Let.YY, and .Yt'2 be Hilbert spaces, and let T:"Y3 Yj C'r= JJt°,. In [XXIII] an operator T:.Y( O+

.JP2 be a bounded operator with .Yt', O+ 'Y3 with C/T=.yY, ED ,k3

is called a dilatation of the operator T if P,T"I .yt', = T" (n = 0, 1, 2,3 , ...), where P, is the orthoprojector from YY, O+ .Y'3 on to .,Yi. Prove that this definition and Definition 3.1 are equivalent when .Y'i =.YW2. 2

Let T and T be the same operators as in Exercise 1. Prove that the operator Twill

be a dilatation of the operator T if and only if P, (T - XI)-" ,W, = (T- XI)-", n = 1, 2, ..., X E p(T) fl p(T) (see [XXIII] ). 3

Suppose that F(X) is a function holomorphic in a neighbourhood of zero with values in a set of continuous operators acting in a J-space Jr', and that there are: a space II, with x negative squares; a continuous operator r :.YP - 11,; a ir-semiunitary operator V -' : II, - II,; and a J-selfadjoint operator S with !Ys = 'Y; such that F(X) = iS+1'`(V+ X1)(V- xI)-'I', I X I < 1, X a(V). Prove that the kernel K(X,µ)= J[F(X)+ F`(µ)] (1 - Xµ) has not more than x negative squares (cf. M. Krein and Langer [4] ).

4

Let T(X) = To: .01 .V2 be an operator-function holomorphic in a neighbourhood of zero. Prove that an operator T :.yt', O+ 'YO - 2 + . Yo will generate the function T(X) if and only if it is a dilatation of the operator To (Azizov).

5

Prove that an operator To admits a x-regular dilatation if and only if To is a (J,, J2)-bi-non-expansive operator (Azizov [14]). Hint: Use the equivalence of assertions a) and b) in Theorem 3.16 and the result of Exercise 4.

6

Give an example of an operator-function T(X) :.yP, - JI2i holomorphic in a neighbourhood of zero, such that the kernel [J, - T*(µ)J2T(X)]/(I - µX) has x negative squares but nevertheless T(X) ri'.

Hint: Put T(X) - To, where To is a (J,, J2)-semi-unitary operator which is not (J,, J2)-bi-non-expansive, and use either the results of Exercises 4 and 5, or the equivalence of assertions a) and b) in Theorem 3.16. 7

Prove that under the conditions of Theorem 3.16 the function T`(X) E II'. Hint: Verify that if the operator T generates the function T(X), then T` generates the function T`(X).

8

Let T(µ):.;o , -..W2 be an operator-function holomorphic in a neighbourhood of the point µ = 0 with values in a set of continuous operators (i T(,) _ .Y',) acting from the J,-space.YY', into the J2-space .W2. Prove that the following assertions are equivalent: a) the kernels

J,-T*(µ)J2T(X) 1 -µX non-negative;

and

J2-T(X)J, T*(µ)

1 -µA

284

5 Theory of Extensions of Isometric and Symmetric Operators b) in a neighbourhood of zero the T(X) are (J,, J2)-bi-non-expansive and the function w' (T(>,)) = T, (X) admits holomorphic extension on to the disc [XI IXI<1)and,IT,(X)II<, 1; Arov[1]). c) the kernel [I, - T,(µ)T,(X)]/l -µX is non-negative O+ + W' is a (J,,J2)-unitary Hint: Verify that if T= 11 T,iII 2 V. operator (J; = J; + I, i = 1, 2) generating the function T(X), then w' (T) generates T, (X), and conversely.

9

Suppose that under the conditions of Theorem 3.16 J; = I;, 1 = 1, 2, and that T(µ) satisfies at least one of the assertions a)-c) of this theorem. Prove that then there is a a,-space II and a (I, v -,, 12 0 Ir,)-unitary operator T= II Ti; II j;_, :./P, O II, - .1P2 b II, generating T(µ) such that T22 has no neutral eigenvalues (Azizov). Hint: Use Theorem 3.18.

10

Under the conditions of Problem 9 suppose J, = I,, J2 = 12, and T(µ) E II,. Prove that then T(µ) admits holomorphic extension on to the disc ( X I I X I < 1) with the

exception, possibly of not more than x points which will be the poles of this extension (M. Krein and Langer [6] ).

Hint: Use the result of Problem 9 and prove that the function T(µ) can be expressed in the form of a product T(µ) = T, (µ)T2(µ), where T, (µ) E II°, and T, (µ) E II' and is generated by a negative 7r,-space II, with dim n, = x. 11

Let T(µ) be an operator-function, holomorphic in a neighbourhood of the point µ = 0, which is a J-bi-extension of a J-isometric operator U for every µ from a certain neighbourhood of the point µ = 0. Prove that among the simple J-unitary operators generating the function T(µ) there is at least one which is an extension of the operator U (Azizov).

Hint: Use the methods of proof of Lemma 3.9, Theorem 3.4 and Lemma 3.8.

Remarks and bibliographic indications on chapter V The definition of the Potapov-Ginzburg transformation in so general a form was introduced essentially by Shmul'yan [5] (cf. Arov [1]). The traditional PG-transformation when A', = ,W'2, J, = J2 was first introduced and studied by Potapov [1] and later Ginzburg [1] continued the investiga§1

tion. The works of I. Iokhvidov [14], [18], E. Iokhvidov and I. Iokhvidov [1], Shmul'yan [4], [5], Ritzner [4] are devoted to the PG-transformation. The exposition in the text, sometimes new in method, is due to Azizov, although almost all the propositions can be found in parts in the works of the authors just mentioned. We mention, in particular, that Theorem 1.14 was obtained independently and simultaneously by Azizov, E. Iokhvidov and I. Iokvidov [1] and Ritsner [4]. §2 The problems of J-isometric extensions of J-isometric operators in the finite-dimensional case were taken up by Potapov [ 1 ] , and in the case of n. by

1. Iokhvidov [15], [16] (see [XIV]; later M. Krein and Shmul'yan [3], E. lokhvidov [2], [4], [5], Azizov [12], [14] studied these problems for operators acting in Krein spaces. In the work of M. Krein and Shmul'yan [3] Jr and .fit, are regular subspaces, and in the articles of E. Iokhvidov [2], [4], [5]

dim V I1] < oo and dim

oo. The latter built his investigations on

Remarks and bibliographical indications on Chapter 5

285

systematic use of PG-transformations (see, e.g., Exercises 5, 6 on §1). With due regard to the above remarks the remaining results in this section and the method used are due to Azizov. §3.1 The concept of a dilatation on operators acting from one space into

another was first,

it

appears, carried over and applied by Azizov

[14]

(Definition 3.1). The remaining results of this paragraph in the case when the operator acts in a single space are well-known (see Davis [1], Davis and Foras [1], Saktinovich [1], Kuzhel' [8], [9]. In our approach these results were proved by Azizov [14], and Corollary 3.5 we find, essentially in Arov [ 1 ] . §3.2 We find the functions T(µ) in Definition 3.6 in Arov [1], for example, where they are called `transfer functions'. Lemma 3.8 and Theorem 3.10 were proved by Azizov [14], and Lemma 3.5 we find, essentially, in Arov [ 1 ] . §3.3 The investigations in this paragraph, carried out by Azizov [14], were stimulated by a series of papers by M. Krein and Langer [3]-[6]. In such a general formulation for Krein spaces neither Theorem 3.12 nor Lemma 3.14 are given by these authors, but our exposition follows their basic ideas, and often repeats their arguments word for word. Lemma 3.15 is due to Azizov. Theorem 3.16 is also due to Azizov [14]. Partially and in particular cases it is encountered in other authors. We single out the series of papers mentioned above of Krein and Langer, and Arov's article, which initiated the appearance of this result. Theorem 3.18 is due to Azizov [14]. §3.4 The set H>. (A) was introduced into the investigation by Azizov. Theorem 3.21 is also due to him (see Azizov [14]). §3.5

Theorem 3.22 is due to Azizov [14]. The form of writing the

generalized resolvent Rx, = (Tx - XI) - ' was first introduced by A Strauss [1].

Moreover we have used M. Krein's idea (see [I]) of all the generalized resolvents by means of a fixed extension of the operator. For more detail about

other approaches to the description of generalized resolvents and historical information see [IV] and also Etkin [1], [2]. Corollary 3.23 is due to Azizov [14], and Corollary 3.24 to A. Shtraus [1]; we point out that the proof in the text, due to Azizov, is not based on A. Shtraus's result.

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INDEX

Adjoint operator 49 Almost J-orthonormalized basis 76 Almost normalized system of vectors 76 Amenable group of operators 137 Angular operator 49 Anti-linearity 2 Anti-space 4 Aperture of two subspaces 61 Asymptotically isotropic sequence 42 B-plus-operator 132

Biorthogonal systems of vectors 76 Bounded spectral set 97

Canonical decomposition of a space 14 Canonical imbedding 183 Canonical imbedding operators 22 Canonical projectors 18 Canonical symmetry operator 21 Carrier a(E) of homomorphism 210 Cauchy-Bunyakovski inequality 6 Cayley-Neyman transformation 142 Centralized system of subsets 59 Class H operators 193 Class K(H) operators 197 Closed mapping in a Hausdorff space 177

Complete system of vectors 219 Completely invariant subspace relative to a family of operators 161 Condition L 160 Condition A+ 170 Conditional basis 77 Continuous spectrum of a linear relation 145

`Corner' 170 Critical points 220

Decomposable lineal 34 Deficiency subspace 25 Definite lineal 3 Definite subspace 24 `Definitizable' J-selfadjoint operator 211 Degenerate lineal 5 Dilatation of an operator 266 Direct Cayley-Neyman transformation 144

Dissipative kernel 234 Dissipative operators 91

Domain of definition of a linear relation 85

Doubly strict plus-operator 120 Dual pair 72

Eigen spectral function of a J-selfadjoint operator 211 Eigenlineal of a linear relation 143 Eigenvalues of a linear relation 143 Eigenvector of L(X) 237 Eigenvectors of a linear relation T 143 Extension of a dual pair 72 Field of regularity of a linear relation 44 Field of regularity of an operator 92 Fixed point of the transformation F; 176

Focusing plus-operator 126 Function generated by an operator 268 Function T(µ) belonging to the class ,r(. j,IY2) 268

G-adjoint of a linear relation 146 G-metric 40 G-selfadjoint linear relation 146 G-selfadjoint operator 104 301

302

Index

G-space 40 G (')-space 69 G-symmetric linear relation 104 G-symmetry 101 (G1, G2)-adjoint 85 (G,, G2)-semi-unitary operator 35

(G,, G2)-unitary operator 135 Generalized linear-fractional transformation 176 Generalized resolvent of a J-symmetric operator 279 Gram operator 39 Hermitian form 1 Hermitian kernel 269 Hermitian non-negative kernel 234 Hermitian operator 104 Hermitian positive kernel 234 Hermitian symmetric 2 Hilbert-Schmidt operator 222 Hyper-maximal neutral lineal 28

J-non-contractive operator 117 J-non-negative 107 J-orthogonal complement 21 J-orthogonal projection 45 J-orthogonality 21 J-orthonormalized bases 72 J-orthonormalized system 74 J-positive 107 J-real part of an operator 96

J-space 20

J-spectral function with a set s(E) of critical points 210 J-symmetry 20 Ji-module 216

(Ji, Jz)-bi-extension of an operator 248 (J1, Jz)-bi-non-contractive operator 124 (J,, Jz)-isometric operator 140 (Ji, Jz)-non-expansive operator 129 (Ji, Jz)-semi-unitary operator 134 (Jl, Jz)-unitary dilatation of an operator 266

(J1, Jz)-unitary operator 141 Identical linear relation 143 Indecomposable lineal 34 Indefinite form 2 Indefinite lineal 3 Indefinite metric 2 Indefiniteness 85 Indefiniteness of a linear relation 143

Jordain chain of associated vectors 237

Kernel 85

Kernel of a linear relation 85 Krein space 14 Krein-Shmul'yan linear-fractional transformation 176

Index of a subspace T, 256 Index of a (D-operator 94 Inertia index 38 Intrinsic completeness 32 Intrinsic metric 31 Intrinsic norm 31 Invariant subset relative to a generalized linear-fractional transformation 176 Invariant subspace 158 Inverse Cayley-Neyman transformation

(t+, 'P_ )-decomposition of a subspace 73

Law of inertia 38 Lineal 2 Linear 1

Linear dimension 24 Linear relation 25 Linear space 1 Linearly ordered subset 7

144

U (.Y°) 51

Involution 19

Isometrical isomorphism 4 Isotropic lineal 5 Isotropic vector 5

J-accumulative linear operator 147 J-bi-extension of a J-Hermitian operator 264 J-complete J-orthonormalized system 74 J-accumulative linear operator 149 J-form 21 J-imaginary part of an operator 96 J-isometricity 20 J-metric 20

Maximal completely invariant subspace 166

Maximal dual pair 72 Maximal extension of a pair 72 Maximal invariant dual pair 166 Maximal invariant subspace 166 Maximal J-accumulative operator 147 Maximal J-orthonormalized system 74 Maximal negative lineal 6 Maximal neutral lineal 6 Maximal neutral subspace 28 Maximal non-degenerate lineal 6 Maximal positive lineal 6

Index

303

Maximal W-accumulative operator 147 Maximal W-dissipative operator 91 Maximal W-symmetric operator 104 Minus-operators 129

Q-orthogonal sets 5 Q-orthogonal vectors 5 (Q1, Q2)-isometric isomorphism 4 (Q,, Q2)-isometrically isomorphic spaces

Negative lineal 3 Negative vector 3 Neutral lineal 3 Neutral vector 3 Non-closed definite lineal 32 Non-decreasing linear operators 155 Non-degenerate lineal 5 Non-negative Hermitian kernel 242 Non-negative lineal 3 Non-negative vector 3 Non-positive lineal 3 Non-positive vector 3 Non-strict plus-operator 118 Normal eigenvalue 89 Normal point 89 Normalized system of vectors 76 Normally decomposable operator 139 Nuclear operator 222

(Qt, Q2)-skew-symmetric isomorphism 4 Quadratic bundle 238 'Quasi-inverse' operator 107

4

Orthogonal canonical orthoprojectors 21 Orthogonal sum Q+ 15 Orthonormalized basis 219 Ortho-projectors 18 P-basis 226 PG-transformation 246 Partial ordering by inclusion 7 Partially (J,, J2)-isometric operators 140 Plus-deficiency 122 Plus operator 117 Point of definite (positive or negative) type 210 Point of regular type 92 Point of spectrum of L(X) 237 Point spectrum vp 143 Point spectrum of a linear relation 143 Pontryagin space H, 64 Positive lineal 3 Positive vector 3 Potapov-Ginzburg transformation 245 Projectionally complete lineal 44 Property (D 165 Property 165 Property 1'J 165 Q-biothogonality 10 Q-metric 7 Q-orthogonal complement 5 Q-orthogonal direct sum 14

Range of indefiniteness 38 Range of values of a linear relation 85 Rank of indefiniteness of a subspace 38

Regular critical point of a J-spectral function 211 Regular definite lineal 34 Regular definite subspace 30 Regular dilatation of an operator 266 Regular extension 262 Regular G-metric 40 Regular generalized resolvent 279 Regular J-spectral function 211 Regular points of a linear relation 143 Regular point of L(X) 237 Regular subspace 46 Residual spectrum of a linear relation 144

Resolution of the identity 36 Resolvent set for a linear relation 143 Riesz basis 77 Riesz projector 97 Root lineal 88 Root vector 88

S-bounded operator 94 S-completely continuous operator 94 S-continuous operator 94 s-number of an operator 222 sp A 223 Y, 222 Scalarly commutative family 206 Selfadjoint operator 16 Semi-bounded below 107 Semi-definite lineal 24 Semi-definite spaces 24 Semi-definite subspace 24 Semi-linearity 2 Sesquilinear form 1 Set of critical points of an operator 211 Simple J-dissipative operator 224 Simple J-unitary operator 278 Single-valued linear relation 143 Singular critical point 211 Singular definite lineal 34 Singular G-metric 40

Index

304

Skew-symmetric isomorphism 4 Skewly linked lineals 10 Spectral function 36 Spectrum a

143

Stable J-unitary operator 137 Stable plus-operator 131 Standard operator 253 Strict plus-operator 118 Strictly dissipative kernel 234 Strictly J-dissipative operator 102 Strongly continuous on the right 36 Strongly damped bundle 241 Strongly stable J-unitary operator 139 Subspaces: dual pair 72 Subspaces: extension of a dual pair 72 Subspaces: maximal dual pair 72 Subspaces: maximal extension of a dual pair 72 Subspaces of the h` class 33 Sum of two linear relations 143 Symmetric operator 104 Trace of an operator 223 Unconditional basis 77 Uniformly definite lineal 30 Uniformly J-dissipative operator 102 Uniformly J-positive 102 Uniformly (J1, J2)-bi-expansive operator 126

Vectors associated with an eigenvector of L(X) 237 W-accumulative linear operator 147 W-accumulative linear relation 147 W-dissipative linear relation 147 W-dissipative operator 91 W-Hermitian operator 104 W-isometric linear relation 146 W-metric 39 W-non-contractive linear relation 146 W-non-expansive linear relation 146 W-space 39

W-symmetric linear relation 147 W-symmetric operator 104 W(')-space 69 (W,, W2)-isometric operator 134 (W1, W2)-non-contractive operator 117 (W,, W2)-semi-unitary operator 134 (W,, W2)-unitary operator 134

x-regular dilatation of an operator 266 x-regular extension 262 x-regular generalized resolvent 279 x-regular Xo-standard extension 264 Xo-standard set of closed J-Hermitian operators 264 a-orthogonal complement 67 a-non-negative kernel 242

Uniformly (J1, J2)-expansive operator 126

Uniformly negative lineal 31 Uniformly positive lineal 31

(D-operator 94 4 o-operator 94 (Do-point of an operator 94