Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

2022/01/31 by Soumitra Ghara, Surjit Kumar, Ghara, Soumitra +5 · 1 citation
Mathematics · #22D10 #46E20 #47A13 #47B32 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2201.13228

openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathcal U(d) be the group of d× d unitary matrices. We find conditions to ensure that a \mathcal U(d)-homogeneous d-tuple \boldsymbol T is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space \mathcal H_K(\mathbb B_d, \mathbb Cⁿ) ⊆ \rm Hol(\mathbb B_d, \mathbb Cⁿ), n= dim ∩_j=1^d ker T^*_j. We describe this class of \mathcal U(d)-homogeneous operators, equivalently, non-negative kernels K quasi-invariant under the action of \mathcal U(d). We classify quasi-invariant kernels K transforming under \mathcal U(d) with two specific choice of multipliers. A crucial ingredient of the proof is that the group SU(d) has exactly two inequivalent irreducible unitary representations of dimension d and none in dimensions 2, … , d-1, d≥ 3. We obtain explicit criterion for boundedness, reducibility and mutual unitary equivalence among these operators.

Cited by

• Weakly \mathcal U(d)-homogeneous commuting tuple of bounded operators

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