9.5.
WEAK INVERTIBILITY AND FACTORIZATION IN CERTAIN SPACES OF
ANALYTIC FUNCTIONS*
is called a symmetric measure if ~...

Author:
Frankfurt R.

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9.5.

WEAK INVERTIBILITY AND FACTORIZATION IN CERTAIN SPACES OF

ANALYTIC FUNCTIONS*

is called a symmetric measure if ~ has the form ~(~,@)=(2X)-'~Y(~)d~ A measure ~ on where w is a finite, positive Borel measure on [0, I], having no mass at 0, and such that and any p, 0 < p < ~, v([r, I]) > 0 for all 0 ~ r < 1. For any function f analytic in ~ we define the generalized mean

0 ~< r <

I.

The class EP(~) consists of all functions f analytic

II# lip, p,= ~

in

~

such that

MpO,;~ ;~) < oo.

In the special case where V is a single unit point mass at I, the means

(2 ) (I) reduce to the

classical H p means, and the EP(~) classes to the standard Hardy classes on ~ . In all cases, EP(~) is isometrically isomorphic to the LP(~)-closure of the polynomials. General properties of these classes are outlined in [I-3]. Numerous investigations of special cases (e.g., the Bergman classes, ~ = area measure) are scattered throughout the literature. A complete bibliography would be quite extensive, and so references here are restricted to those which have had the most direct influence upon the author's work. A function f, f ~ EP(~), is said to be weakly invertible if there is a sequence of polynomials {Pn} such that pn f § ~ in the metric of EP(~). From an operator-theoretic point of view, such functions are significant in that an element of E P ( ~ ) i s weakly invertible if and only if it is a cyclic vector for the operator of multiplication by z on EP(~). (When p = 2, this operator is unitarily equivalent to a subnormal weighted shift.) In the special case of the Hardy classes, Beurling [4] showed that a function is weakly invertible if and only if it is outer. In the more general context of the EP(~) classes, a complete characterization of the weakly invertible functions awaits discovery. As this juncture, however, it is not even clear what general shape such a characterization might take. We know of only a handful of scattered results which are applicable to these special classes. The earliest of these can be found in three papers by Shapiro [5-7] and in the survey article by Mergelyan [8]. More recent contributions have been made by the author [I, 2], Aharonov, Shapiro, and Shields [9], and Hedberg (see [10, p. 112]). Many of the known results on weakly invertible functions in the EP(~) classes are essentially either multiplication or factorization theorems. It is well known that the product of two outer functions is outer, and that any factor of an outer function is outer. Do these properties carry over to weakly invertible functions in the EP(~) classes? We list a number of specific questions along these lines. (a) Suppose f, g, h E EP(~) and f = gh. If g and h are weakly invertible, is f weakly invertible? Conversely, if f is weakly invertible, are g and h weakly invertible? (b) If f ~ EP(~) vertible?

is nonvanishing and I/f ~

(c) If f ~

is weakly

EP(~)

Eq(p) for some q, q > 0, is f weakly in-

invertible and a > 0, is f~ weakly invertible

(d) If f ~ EP(~) and f is weakly invertible invertible in EP(~)?

in EP/~(~)?

in Eq(~) for some q, q < p, is f weakly

(e) Let f E EP(~), g E Eq(p), and h 6 ES(~), and let f = gh. If g and h are weakly invertible in Eq(~) and ES(~), respectively, is f weakly invertible in EP(~)? What about the converse? *RICHARD FRANKFURT. Department of Mathematics, Kentucky, Lexington 40506.

College of Arts and Sciences, University of

2189

Of course, all Question (e) is the affirmative answers affirmative answers

these things are trivially true in the special case of the Hardy classes. most general of the list. The reader can easily convince himself that to (e) would imply affirmative answers to all the others. Conversely, to (a) and (d) together would yield affirmative answers to (e).

The answers to question (a) are known to be affirmative if h ~ H ~, or if g 6 EP'(u) and h ~ Eq'(~) with I/p' + I/q' = I/p (see [2]).* These results were inspired by an earlier result of Shapiro [5, Lemma 2]. The question remains unanswered for unrestricted g and h. Question (b) has a long history, and versions of it appear in numerous sources. An affirmative answer may be obtained by imposing the additional condition f ~ EP+6(D) for some ~, 6 > 0. The legacy of results of this type seems to begin with the paper of Shapiro [6], and has been carried forth into a variety of different settings in the separate researches of Brennan [11], Hedberg [12], and the author [2]. A similar result with a different kind of side condition is to be found in the work of Aharonov, Shapiro, and Shields [9]. In its full generality, however, the question remains unanswered. Question (d) seems in some sense to be the crucial question, certainly in moving from the setting of question (a) to that of question (e), but perhaps also in removing the side conditions from the results cited above. Presently, however, there seems to be little evidence either for or against an affLrmative answer, nor can we offer any tangible ideas on how to attack the problem. The key to its solution in the special case of the Hardy classes rests upon the fact that, there, weak invertibility can be accounted for in terms of behavior within the larger Nevanlinna class. Unfortunately, in the more general setting of the EP(~) classes, none of the several different generalizations of the Nevanlinna theory discovered to date seems to shed any light upon the matter. It may very well be that the answer to the question is negative. Clearly, a negative answer would introduce complications which have no parallel in the Hardy classes. However, in view of the negative results of Horowitz [13] concerning the zero sets of functions in the Bergman classes, such complications would not be too surprising, and perhaps not altogether unwelcome. LITERATURE CITED I. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

R. Frankfurt, "Subnormal weighted shifts and related function spaces," J. Math. Anal. Appl., 52, 471-489 (1975). R. Frankfurt, "Subnormal weighted shifts and related function spaces. II," J. Math. Anal. Appl., 55, 1-17 (1976). R. Frankfurt, "Function spaces associated with radially symmetric measures," J. Math. Anal. Appl., 60, 502-541 (1977). A. Beurling, "On two problems concerning linear transformations in Hilbert space," Acta Math., 81, 239-255 (1949). H. S. Shapiro, "Weakly invertible elements in certain function spaces, and generators of ~i," Mich. Math. J., 11, 161-165 (1964). H. S. Shapiro, "Weighted polynomial approximation and boundary behavior of analytic functions," in: Contemporary Problems in the Theory of Analytic Functions, Nauka, Moscow (1966), pp. 326-335. G. Shapiro (H. S. Shapiro), "Some observations concerning the weights polynomial approximation of holomorphic functions," Mat. Sb., 73, No. 3, 320-330 (1967). S. N. Mergelyan, "On the completeness of systems of analytic functions," Usp. Mat. Nauk, 8, No. 4, 3-63 (1953). D. Aharonov, H. S. Shapiro, and A. L. Shields, "Weakly invertible elements in the space of square-summable holomorphic functions," J. London Math. Soc., No. 9, 183-192 (1974). A. L. Shields, "Weighted shift operators and analytic function theory," in: Topics in Operator Theory, Amer. Math. Soc., Providence (1974), pp. 49-128. J. Brennan, "Invariant subspaces and weighted polynomial approximation," Ark. Mat., No.

II, 12. 13.

167-189 (1973).

L. I. Hedberg, "Weighted mean approximation in Carath4odory regions," Math. Scand., 23, 113-122 (1968). C. Horowitz, "Zeros of functions in the Bergman spaces," Duke Math. J., 4__11,693-710

(1974). *See also [3] of 7.5 -- Ed.

2190

WEAK INVERTIBILITY AND FACTORIZATION IN CERTAIN SPACES OF

ANALYTIC FUNCTIONS*

is called a symmetric measure if ~ has the form ~(~,@)=(2X)-'~Y(~)d~ A measure ~ on where w is a finite, positive Borel measure on [0, I], having no mass at 0, and such that and any p, 0 < p < ~, v([r, I]) > 0 for all 0 ~ r < 1. For any function f analytic in ~ we define the generalized mean

0 ~< r <

I.

The class EP(~) consists of all functions f analytic

II# lip, p,= ~

in

~

such that

MpO,;~ ;~) < oo.

In the special case where V is a single unit point mass at I, the means

(2 ) (I) reduce to the

classical H p means, and the EP(~) classes to the standard Hardy classes on ~ . In all cases, EP(~) is isometrically isomorphic to the LP(~)-closure of the polynomials. General properties of these classes are outlined in [I-3]. Numerous investigations of special cases (e.g., the Bergman classes, ~ = area measure) are scattered throughout the literature. A complete bibliography would be quite extensive, and so references here are restricted to those which have had the most direct influence upon the author's work. A function f, f ~ EP(~), is said to be weakly invertible if there is a sequence of polynomials {Pn} such that pn f § ~ in the metric of EP(~). From an operator-theoretic point of view, such functions are significant in that an element of E P ( ~ ) i s weakly invertible if and only if it is a cyclic vector for the operator of multiplication by z on EP(~). (When p = 2, this operator is unitarily equivalent to a subnormal weighted shift.) In the special case of the Hardy classes, Beurling [4] showed that a function is weakly invertible if and only if it is outer. In the more general context of the EP(~) classes, a complete characterization of the weakly invertible functions awaits discovery. As this juncture, however, it is not even clear what general shape such a characterization might take. We know of only a handful of scattered results which are applicable to these special classes. The earliest of these can be found in three papers by Shapiro [5-7] and in the survey article by Mergelyan [8]. More recent contributions have been made by the author [I, 2], Aharonov, Shapiro, and Shields [9], and Hedberg (see [10, p. 112]). Many of the known results on weakly invertible functions in the EP(~) classes are essentially either multiplication or factorization theorems. It is well known that the product of two outer functions is outer, and that any factor of an outer function is outer. Do these properties carry over to weakly invertible functions in the EP(~) classes? We list a number of specific questions along these lines. (a) Suppose f, g, h E EP(~) and f = gh. If g and h are weakly invertible, is f weakly invertible? Conversely, if f is weakly invertible, are g and h weakly invertible? (b) If f ~ EP(~) vertible?

is nonvanishing and I/f ~

(c) If f ~

is weakly

EP(~)

Eq(p) for some q, q > 0, is f weakly in-

invertible and a > 0, is f~ weakly invertible

(d) If f ~ EP(~) and f is weakly invertible invertible in EP(~)?

in EP/~(~)?

in Eq(~) for some q, q < p, is f weakly

(e) Let f E EP(~), g E Eq(p), and h 6 ES(~), and let f = gh. If g and h are weakly invertible in Eq(~) and ES(~), respectively, is f weakly invertible in EP(~)? What about the converse? *RICHARD FRANKFURT. Department of Mathematics, Kentucky, Lexington 40506.

College of Arts and Sciences, University of

2189

Of course, all Question (e) is the affirmative answers affirmative answers

these things are trivially true in the special case of the Hardy classes. most general of the list. The reader can easily convince himself that to (e) would imply affirmative answers to all the others. Conversely, to (a) and (d) together would yield affirmative answers to (e).

The answers to question (a) are known to be affirmative if h ~ H ~, or if g 6 EP'(u) and h ~ Eq'(~) with I/p' + I/q' = I/p (see [2]).* These results were inspired by an earlier result of Shapiro [5, Lemma 2]. The question remains unanswered for unrestricted g and h. Question (b) has a long history, and versions of it appear in numerous sources. An affirmative answer may be obtained by imposing the additional condition f ~ EP+6(D) for some ~, 6 > 0. The legacy of results of this type seems to begin with the paper of Shapiro [6], and has been carried forth into a variety of different settings in the separate researches of Brennan [11], Hedberg [12], and the author [2]. A similar result with a different kind of side condition is to be found in the work of Aharonov, Shapiro, and Shields [9]. In its full generality, however, the question remains unanswered. Question (d) seems in some sense to be the crucial question, certainly in moving from the setting of question (a) to that of question (e), but perhaps also in removing the side conditions from the results cited above. Presently, however, there seems to be little evidence either for or against an affLrmative answer, nor can we offer any tangible ideas on how to attack the problem. The key to its solution in the special case of the Hardy classes rests upon the fact that, there, weak invertibility can be accounted for in terms of behavior within the larger Nevanlinna class. Unfortunately, in the more general setting of the EP(~) classes, none of the several different generalizations of the Nevanlinna theory discovered to date seems to shed any light upon the matter. It may very well be that the answer to the question is negative. Clearly, a negative answer would introduce complications which have no parallel in the Hardy classes. However, in view of the negative results of Horowitz [13] concerning the zero sets of functions in the Bergman classes, such complications would not be too surprising, and perhaps not altogether unwelcome. LITERATURE CITED I. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

R. Frankfurt, "Subnormal weighted shifts and related function spaces," J. Math. Anal. Appl., 52, 471-489 (1975). R. Frankfurt, "Subnormal weighted shifts and related function spaces. II," J. Math. Anal. Appl., 55, 1-17 (1976). R. Frankfurt, "Function spaces associated with radially symmetric measures," J. Math. Anal. Appl., 60, 502-541 (1977). A. Beurling, "On two problems concerning linear transformations in Hilbert space," Acta Math., 81, 239-255 (1949). H. S. Shapiro, "Weakly invertible elements in certain function spaces, and generators of ~i," Mich. Math. J., 11, 161-165 (1964). H. S. Shapiro, "Weighted polynomial approximation and boundary behavior of analytic functions," in: Contemporary Problems in the Theory of Analytic Functions, Nauka, Moscow (1966), pp. 326-335. G. Shapiro (H. S. Shapiro), "Some observations concerning the weights polynomial approximation of holomorphic functions," Mat. Sb., 73, No. 3, 320-330 (1967). S. N. Mergelyan, "On the completeness of systems of analytic functions," Usp. Mat. Nauk, 8, No. 4, 3-63 (1953). D. Aharonov, H. S. Shapiro, and A. L. Shields, "Weakly invertible elements in the space of square-summable holomorphic functions," J. London Math. Soc., No. 9, 183-192 (1974). A. L. Shields, "Weighted shift operators and analytic function theory," in: Topics in Operator Theory, Amer. Math. Soc., Providence (1974), pp. 49-128. J. Brennan, "Invariant subspaces and weighted polynomial approximation," Ark. Mat., No.

II, 12. 13.

167-189 (1973).

L. I. Hedberg, "Weighted mean approximation in Carath4odory regions," Math. Scand., 23, 113-122 (1968). C. Horowitz, "Zeros of functions in the Bergman spaces," Duke Math. J., 4__11,693-710

(1974). *See also [3] of 7.5 -- Ed.

2190

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