A Trace Inequality for Commuting d-Tuples of Operators

Misra, G and Pramanick, P and Sinha, KB (2022) A Trace Inequality for Commuting d-Tuples of Operators. In: Integral Equations and Operator Theory, 94 (2).

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Abstract

For a commuting d-tuple of operators T defined on a complex separable Hilbert space H, let [[T∗,T]] be the d× d block operator (([Tj∗,Ti])) of the commutators [Tj∗,Ti]:=Tj∗Ti-TiTj∗. We define the determinant of [[T∗,T]] by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of [[T∗,T]] equals the generalized commutator of the 2d - tuple of operators, (T1,T1∗,…,Td,Td∗) introduced earlier by Helton and Howe. We then apply the Amitsur–Levitzki theorem to conclude that for any commuting d-tuple of d-normal operators, the determinant of [[T∗,T]] must be 0. We show that if the d-tuple T is cyclic, the determinant of [[T∗,T]] is non-negative and the compression of a fixed set of words in Tj∗ and Ti—to a nested sequence of finite dimensional subspaces increasing to H—does not grow very rapidly, then the trace of the determinant of the operator [[T∗,T]] is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting d-tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Item Type:	Journal Article
Publication:	Integral Equations and Operator Theory
Publisher:	Birkhauser
Additional Information:	The copyright for this article belongs to Birkhauser
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	19 May 2022 04:55
Last Modified:	19 May 2022 04:55
URI:	https://eprints.iisc.ac.in/id/eprint/72030

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