Tridiagonal reproducing kernels and subnormality

Adams, Gregory T.; Feldman, Nathan S.; McGuire, Paul J.

doi:10.7900/jot.2011sep12.1942

Cited by 3 publications

(14 citation statements)

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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%

“…Let T 2 and S λ be as discussed in Example 4.1. Let x := e (0,0) , y := λ (2,1) e (1,1) − λ (1,1) e (2,1) . Assume that the weight sequence {λ (j,i) : i ≥ 1, j = 1, 2} of S λ satisfies the following: (2,2) for every integer n ≥ 3.…”

Section: Examplesmentioning

confidence: 99%

“…Example 4.3. The preceding proposition is applicable to S λ with weights λ (1,1) (1,3) , and λ (2,3) = 1 = λ (j,i) for i ≥ 4 and for j = 1, 2. In this case, α = 1 8 , α 1 = 1 2 , α 2 = 3 8 and α k = 1 8 for all integers k ≥ 3.…”

Section: Examplesmentioning

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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Section: Examplesmentioning

confidence: 67%

Section: Preliminariesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

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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“…It is well-known that a sequence {a k } k∈Z + is completely monotone if and only if the sequence {a k } k∈Z + is a Hausdorff moment sequence, that is, there exists a positive measure ν supported in [0, 1] such that a k = [0, 1] x k dν(x) for all k ∈ Z + (cf. [9]).…”

Section: Introductionmentioning

confidence: 99%

On the sum of two subnormal kernels

Ghara

Kumar

2019

Journal of Mathematical Analysis and Applications

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We show, by means of a class of examples, that if K1 and K2 are two positive definite kernels on the unit disc such that the multiplication by the coordinate function on the corresponding reproducing kernel Hilbert space is subnormal, then the multiplication operator on the Hilbert space determined by their sum K1 + K2 need not be subnormal. This settles a recent conjecture of Gregory T. Adams, Nathan S. Feldman and Paul J. McGuire in the negative. We also discuss some cases for which the answer is affirmative.

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Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms

Das

Sarkar

2023

Rev. Mat. Iberoam.

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Tridiagonal reproducing kernels and subnormality

Cited by 3 publications

References 7 publications

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

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

On the sum of two subnormal kernels

Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms

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