A class of tridiagonal reproducing kernels

Adams, Gregory T.; McGuire, Paul J.

doi:10.7153/oam-02-15

Cited by 2 publications

(10 citation statements)

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“…These results generalize those in [3] and [9] to higher bandwidth and more general weight sequences. This leads to a very nice functional characterization of certain finite bandwidth spaces.…”

Section: The Problemsupporting

confidence: 84%

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A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

Adams¹,

Wagner²

2019

Preprint

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In this work, we consider "finite bandwidth" reproducing kernel Hilbert spaces which have orthonormal bases of the form f n (z) = z n J j=1 (1 − a n w j z), where w 1 , w 2 , . . . w J are distinct points on the circle T and {a n } is a sequence of complex numbers with limit 1. We provide general conditions based on a matrix recursion that guarantee such spaces contain a functional multiple of the Hardy space. Then we apply this general method to obtain strong results for finite bandwidth spaces when lim n→∞ n(1 − a n ) = p. In particular, we show that point evaluation can be extended boundedly to precisely J additional points on T and we obtain an explicit functional decomposition of these spaces for p > 1/2 in analogy with a previous result in the tridiagonal case due to Adams and McGuire. We also prove that multiplication by z is a bounded operator on these spaces and that they contain the polynomials.

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Section: The Problemsupporting

confidence: 84%

“…Adams and McGuire established that the natural domain for H(K) is a disk about the origin with up to J additional points [2]. They explored the J = 1 case and gave an interesting family of kernels K where H(K) is a nontrivial extension of a diagonal space [3]. In this paper, we show how to generalize their results to higher bandwidths.…”

Section: The Problemmentioning

confidence: 79%

A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

Adams¹,

Wagner²

2019

Preprint

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“…Let S λ be a left-invertible weighted shift on T 3 . As in the preceding example, one can see that E := ker S * λ = {αe (0,0) + β(λ (3,1) e (2,1) − λ (2,1) e (3,1) ) : α, β ∈ C}.…”

Section: Examplesmentioning

confidence: 67%

“…Although the motivation for the present work comes mainly from the theory of weighted shifts on directed trees as expounded in [20], it is closely related to some of the recent developments in the function theoretic operator theory. In particular, the reader is referred to the study of analytic reproducing kernels of finite bandwidth carried out in a series of papers by G. Adams et al [2], [3], [1], [4] (refer also to [6] for the general theory of reproducing kernels). It is also worth noting that the class of weighted shifts on rooted directed trees has some resemblance with the class of adjoints of abstract weighted shifts (in the context of complex Hilbert spaces) [7], [8], [26] and also with the class of operator-valued weighted shifts [25], [23], [18], [30] studied extensively in the literature.…”

Section: Preliminariesmentioning

confidence: 99%

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An analytic model for left-invertible weighted shifts on directed trees: Table 1.

Chavan¹,

Trivedi²

2016

J. London Math. Soc.

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Let T be a rooted directed tree with finite branching index k T , and let S λ ∈ B(l 2 (V )) be a leftinvertible weighted shift on T . We show that S λ can be modelled as a multiplication operator Mz on a reproducing kernel Hilbert space H of E-valued holomorphic functions on a disc centred at the origin, where E := ker S * λ . The reproducing kernel associated with H is multi-diagonal and of bandwidth k T . Moreover, H admits an orthonormal basis consisting of polynomials in z with at most k T + 1 non-zero coefficients. As one of the applications of this model, we give a spectral picture of S λ . Unlike the case dim E = 1, the approximate point spectrum of S λ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed tree with finite branching index.Let H be a complex separable Hilbert space. The inner-product on H will be denoted by ·, · H . If no confusion is likely, then we suppress the suffix, and simply write the inner-product as ·, · . By a subspace, we mean a closed linear manifold. Let W be a subset of H. Then span W stands for the smallest linear manifold generated by W. In case W is singleton {w}, we use the convenient notation w in place of span {w}. By {w : w ∈ W }, we understand the subspace generated by W . For a subspace M of H, we use P M to denote the orthogonal projection of H onto M. For vectors x, y ∈ H, we use the notation x ⊗ y to denote the rank 1 operator given byUnless stated otherwise, all the Hilbert spaces occurring below are complex infinitedimensional separable and, for any such Hilbert space H, by B(H) we denote the Banach algebra of bounded linear operators on H. For T ∈ B(H), the symbols ker T and ran T will stand for the kernel and the range of T, respectively. The Hilbert space adjoint of T will be denoted by T * . In what follows, we denote the spectrum, approximate point spectrum, essential spectrum and the point spectrum of T by σ(T ), σ ap (T ), σ e (T ) and σ p (T ), respectively. We reserve the notation r(T ) for the spectral radius of T.Let T ∈ B(H). We say that T is left-invertible if there exists S ∈ B(H) such that ST = I. Note that T is left-invertible if and only if there exists a constant α > 0 such that T * T αI. In this case, T * T is invertible and T admits the left-inverse (T * T ) −1 T * . Following [30], we refer to the operator T given by T := T (T * T ) −1 as the Cauchy dual of the left-invertible operator T. Further, we say that T is analytic if n 0 T n (H) = {0}. If H is a reproducing kernel Hilbert space of holomorphic functions defined on a disc in C, then the multiplication operator M z defined on H provides an example of an analytic operator. It is interesting to note that almost all analytic operators arise in this way. Indeed, a result of Shimorin [30] asserts that any left-invertible analytic operator is unitarily equivalent to the operator of multiplication by z on a reproducing kernel Hilbert space of vector-valued holomorphic functions

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A class of tridiagonal reproducing kernels

Abstract: Abstract. The class of analytic reproducing kernels

Cited by 2 publications

References 10 publications

A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

An analytic model for left-invertible weighted shifts on directed trees: Table 1.

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