An F. and M. Riesz Theorem for strong H2-functions on the unit circle
Introduction
Let be the Hardy space of the unit circle in the complex plane . An F. and M. Riesz Theorem states that if , then (cf. [4]). However, the statement (1) is liable to fail for operator-valued functions because they have zero divisors even though f and g are matrix-valued functions. Thus we may ask under what condition on operator-valued functions f and g does the statement (1) hold? In this paper, we extend the statement (1) for operator-valued functions.
We first review a few essential facts concerning vector-valued - and -functions, using [3], [6], [7], [8], and [10] for general references. For a complex Banach space X and , let be the set of all strongly (Lebesgue) measurable functions such that For , the n-th Fourier coefficient of f, denoted by , is defined by where the integral is understood in the sense of the Bochner integral. For , define If E is a complex Hilbert space then is a complex Hilbert space with the inner product defined by Let be the open unit disk in . For a complex Banach space X, denote for the set of all functions analytic on satisfying Throughout the paper, suppose D and E are infinite-dimensional separable complex Hilbert spaces. As in the scalar-valued case, if , then there exists a “boundary function” such that where denotes the Poisson integral. The Taylor coefficients of f coincide with the Fourier coefficients of bf, and a.e. Moreover, the mapping is an isometric bijection from onto (cf. [6, Theorem 3.11.7]). Thus if , then there exists such that . In this case, we write .
Recall that a function is called a strong -function if for each (cf. [5]). Write for the space of -valued strong -functions on . On the other hand, for , define the class by the set of all WOT measurable -valued functions Φ on satisfying Following to V. Peller [7], a function is called a strong -function.
For , a Toeplitz operator is a densely defined operator defined by where denotes the orthogonal projection from onto and is the set of all polynomials with values in D. Very recently, a systematic study on strong -functions was undertaken in the study of -invariant subspaces (see [2]), where is a shift operator on defined by for each . If and , then . We now define the n-th Fourier coefficient of , denoted by , by It is easy to show that is linear transformation from D to E for each . We define or equivalently, is the set of all WOT measurable functions Φ on such that for each . A function is called a strong -function on the unit circle . We can easily check that
In this paer, we establish an extension of the statement (1) for strong -functions on the unit circle .
For and , write
Our main theorem now states:
Theorem 1.1 Let and . Suppose the 0-th Fourier coefficient of has dense range for some . If a.e. on , then a.e. on .
Let denote the set of complex matrices and write for the set of matrix-valued -functions, i.e., We can easily check that .
As a corollary of Theorem 1.1, we then have:
Corollary 1.2 Let and . Suppose or is injective for some . If a.e. on , then either a.e. on or a.e. on .
A significance of the Corollary 1.2 lies in the fact that some condition on the front factor or the back factor of the product holds for at least one point in the open unit disk then an operator-valued version of the F. and M. Riesz Theorem (1) holds.
To illustrate Corollary 1.2, consider Clearly, . Observe is invertible. Thus by Corollary 1.2, for any nonzero ().
In Section 2, we give proofs of Theorem 1.1 and Corollary 1.2.
Section snippets
Proofs of the main results
To prove Theorem 1.1 and Corollary 1.2, we need several auxiliary lemmas. We begin with recalling the Poisson kernel defined by If for a complex Banach space X, let denote the Poisson integral of f defined by For , let for . Then we can see that Thus maps sets of measure 0 to sets of measure 0 and in turn, maps measurable sets
Acknowledgments
The authors are indebted to the referee for many helpful suggestions. The work of the second named author was supported by NRF (Korea) grant No. 2019R1A2C1005182. The work of the third named author was supported by NRF (Korea) grant No. 2019R1A6A1A10073079. The work of the fourth named author was supported by NRF (Korea) grant No. 2018R1A2B6004116.
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