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

Kumar, Surjit

doi:10.1007/s11785-015-0462-y

Cited by 7 publications

(5 citation statements)

References 15 publications

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“…In this section, we study the question of unitary equivalence and similarity of two commuting d-tuple of operators in the class AK(Ω). In particular, when K is the unit circle T, these results were obtained by Shields in [27] and the case when K is U(d), the similarity result was obtained in [19,Lemma 2.2]. By Theorem 2.3, any d-tuple of operators T in AK(Ω) is unitarily equivalent to M (a) consisting of multiplication operators by the coordinate functions z 1 , .…”

Section: Conjecture 46supporting

confidence: 57%

$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

2021

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Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T 1 , . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z 1 , . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B 1 (Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i T i is given. In general, based on this formula, we make a conjecture giving the form of this operator.

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Section: Conjecture 46supporting

confidence: 57%

$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

2021

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“…In the particular case when K is the unit circle T, these results were obtained by Shields in [27]. The higher-dimensional counterpart of similarity result is obtained in [19,Lemma 2.2].…”

Section: Unitary Equivalence and Similaritymentioning

confidence: 63%

$\mathbb K$-homogeneous tuple of operators on bounded symmetric domains

Ghara¹,

Kumar²,

Pramanick³

2020

Preprint

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Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T1, . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for all K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For a bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i Ti is given. In general, based on this formula, we make a conjecture giving the form of this operator.

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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%

“…The first of which is a local spherical analog of von Neumann's inequality (cf. [75,Proposition 2.5] and [71,Theorem 7.6]).…”

Section: Spherically Balanced Multishiftsmentioning

confidence: 99%