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