Sum of self-commutators of commuting operators
Abstract
It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal d-tuples on a complex Hilbert space ℋ, namely, commuting d-tuples bf T=(T1, …, Td) satisfying ∑j=1d [T*j, Tj] geqslant 0. We show that every sum-hyponormal d-tuple of compact operators decomposes as the direct sum of a normal d-tuple and a quasinilpotent sum-hyponormal d-tuple. As a consequence, every sum-hyponormal d-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal d-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal d-tuples with commuting imaginary parts. As a consequence, we prove that bf T is normal whenever the planar Lebesgue measure of s(σ(bf T)) is zero, where σ(bf T) denotes the Taylor spectrum of bf T and s is the complex-linear polynomial with alternating coefficients $1$ and i.
Related
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Commuting Toeplitz operators with harmonic symbols on the Fock space
- Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space
- Essentially semi-commuting singular integral operators with Cauchy kernel on L2
- Applications of the Projection Characterization and Symmetry Bootstrap for Operators that are Nearby Commuting Operators
BibTeX
@misc{chavan2026selfcommutators,
title = {{Sum of self-commutators of commuting operators}},
author = {Sameer Chavan and Md. Ramiz Reza and Shanola S. Sequeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19287},
eprint = {2609.19287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19287},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sum of self-commutators of commuting operators
AU - Sameer Chavan
AU - Md. Ramiz Reza
AU - Shanola S. Sequeira
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19287
UR - https://arxiv.org/abs/2609.19287
AB - It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal d-tuples on a complex Hilbert space ℋ, namely, commuting d-tuples bf T=(T_1, …, T_d) satisfying ∑_j=1^d [T^*_j, T_j] geqslant 0. We show that every sum-hyponormal d-tuple of compact operators decomposes as the direct sum of a normal d-tuple and a quasinilpotent sum-hyponormal d-tuple. As a consequence, every sum-hyponormal d-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal d-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal d-tuples with commuting imaginary parts. As a consequence, we prove that bf T is normal whenever the planar Lebesgue measure of s(σ(bf T)) is zero, where σ(bf T) denotes the Taylor spectrum of bf T and s is the complex-linear polynomial with alternating coefficients $1$ and i.
ER -