# A Trace Inequality for Commuting d-Tuples of Operators

## Article Type

Research Article

## Publication Title

Integral Equations and Operator Theory

## 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.

## DOI

10.1007/s00020-022-02693-5

## Publication Date

6-1-2022

## Recommended Citation

Misra, Gadadhar; Pramanick, Paramita; and Sinha, Kalyan B., "A Trace Inequality for Commuting d-Tuples of Operators" (2022). *Journal Articles*. 3098.

https://digitalcommons.isical.ac.in/journal-articles/3098