Spherically balanced Hilbert spaces of formal power series in several variables

Chavan, Sameer; Kumar, Surjit

doi:10.7900/jot.2013apr22.2000

Cited by 7 publications

(7 citation statements)

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“…, k and hence γ t p ε p +ε q = γ t p ε p +ε q for all 1 ≤ p, q ≤ k. It now follows that γ t = γ t for all t ∈ Z k + . Now μ = ν follows by imitating the proof of second half of [8,Lemma 3.4]. It is now easy to see that δ n, p δ n+ε i ,q = γ t+ε p γ t .…”

Section: Remark 33mentioning

confidence: 85%

Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

Kumar

2015

Complex Anal. Oper. Theory

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We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space H 2 (β) of formal power series in the variables z 1 , . . . , z m is spherically balanced if and only if there exist a Reinhardt measure μ supported on the unit sphere ∂B and a Hilbert space H 2 (γ ) of formal power series in the variable t such thatwhere f z (t) = f (tz) is a formal power series in the variable t. In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple M z on a spherically balanced space H 2 (β). We also observe that all spherical contractive multishifts on spherically balanced spaces admit the classical von Neumann's inequality.In the second half, we introduce and study a class of Hilbert spaces, to be referred to as G-balanced Hilbert spaces, where G = U(r 1 ) × U(r 2 ) × · · · × U(r k ) is a subgroup of U(m) with r 1 + · · · + r k = m. In the case in which G = U(m), G-balanced spaces are precisely spherically balanced Hilbert spaces.

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Section: Remark 33mentioning

confidence: 85%

Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

Kumar

2015

Complex Anal. Oper. Theory

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“…Here, by the Reinhardt measure, we mean a T d -invariant finite positive Borel measure supported in ∂B, where T d denotes the the unit d-torus {z ∈ C d : |z 1 | = 1, · · · , |z d | = 1}. For more details on spherically balanced Hilbert spaces, we refer to [11].…”

Section: Multi-variable Casementioning

confidence: 99%

“…The following lemma has been already recorded in [11,Lemma 4.3]. We include a statement for ready reference.…”

Section: Multi-variable Casementioning

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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“…In this chapter, we discuss two classes of so-called balanced multishifts, namely torally balanced multishifts and spherically balanced multishifts (cf. [31], [75]). These generalize largely the classes of toral and spherical isometries ( [18], [50], [53], [3]).…”

Section: Special Classes Of Multishiftsmentioning

confidence: 99%