Analytic reproducing kernels and multiplication operators

Adams, Gregory T.; McGuire, Paul J.; Paulsen, Vern I.

doi:10.1215/ijm/1255987417

Cited by 23 publications

(17 citation statements)

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“…If K(z, w) is an analytic reproducing kernel in a neighborhood of (0, 0), then K has a power series expansion a ij z iwj about (0, 0), and the reproducing kernel Hilbert space obtained by restricting K to the connected domain containing (0, 0) is denoted by H(A). In a paper by Adams, McGuire and Paulsen [1], it is shown that the coefficient matrix A = [a i,j ] is a positive definite matrix in l 2 , and that H(A) is isomorphic to R(B), where A = BB * and R(B) is the range space of B equipped with the norm Bx = y , where y is the unique vector in (ker B) ⊥ such that By = Bx. The isomorphism associates the function f(z) = ∞ n=0 α n z n with the vector (α 0 , α 1 , .…”

Section: Examplementioning

confidence: 99%

“…Since such a space would arise as H(DD * ), where D is a diagonal matrix, we are asking when H(LL * ) = H(DD * ) as a set of functions. This is equivalent to Range (L) = Range (D), where Range (L) is the range space (see Adams, McGuire and Paulsen [1]). By Douglas [3], Range (L) = Range (D) is equivalent to the existence of bounded matrices C 1 and C 2 such that LC 1 = D and DC 2 = L. Thus LC 1 C 2 = L. As L is one-to-one, C 1 C 2 = I.…”

Section: Tridiagonal Kernel Spacesmentioning

confidence: 99%

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Analytic Tridiagonal Reproducing Kernels

Adams

McGuire

2001

J. Lond. Math. Soc.

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Abstract. This paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(a n,0 + a n,1 z + · · · + a n,J z J )z n , n ≥ 0}.The primary focus is on the tridiagonal case where J = 1 and how it compares to the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated and aspects of this operator are discussed.In the diagonal case Mz is a weighted unilateral shift. It is shown that in the tridiagonal case this need not be so and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.

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Section: Examplementioning

confidence: 99%

Section: Tridiagonal Kernel Spacesmentioning

confidence: 99%

Analytic Tridiagonal Reproducing Kernels

Adams

McGuire

2001

J. Lond. Math. Soc.

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“…Hence M * z satisfies the Hypercyclicity Criterion with respect to {n k }. This proves (1). For the second part, we proceed with D and f as in the proof of part (1) above.…”

Section: Dynamics Of M * Zmentioning

confidence: 79%

Linear dynamics in reproducing kernel Hilbert spaces

Mundayadan

Sarkar

2020

Bulletin des Sciences Mathématiques

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Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for M * z , the adjoint of M z , on vector-valued analytic reproducing kernel Hilbert spaces H in terms of the derivatives of kernel functions on the open unit disc D in C. Here M z denotes the multiplication operator by the coordinate function z, that is, for all f ∈ H and w ∈ D. We analyze the special case of quasi-scalar reproducing kernel Hilbert spaces. We also present a complete characterization of hypercyclicity of M * z on tridiagonal reproducing kernel Hilbert spaces and some special classes of vector-valued analytic reproducing kernel Hilbert spaces.∂ 2n k ∂z n ∂w n (0, 0) = 0.

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“…For a general reference on these spaces see either [5] or [11]. We use the set up from [1] and [2]. Let C be a Hilbert space and G ⊆ C d be an open neighborhood of 0.…”

Section: Factorization and Multi-variable Reproducing Kernel Hilbert mentioning

confidence: 99%

“…,z d ) and z I = z i 1 1 · · · z i d d . If Q = (Q I,J ) I,J≥0 , Q I,J ∈ B(C), is positive semidefinite on finite sections and K(z, w) := I,J≥0 z IwJ Q I,J is convergent on some polydisk, then by results of[1], K(z, w) is positive on that polydisk.…”

mentioning

confidence: 99%