2020/12/21 by Gadadhar Misra, Misra, Gadadhar, Paramita Pramanick +3
Mathematics · #(2020) Primary: 47B10 #47A16 #47B20 Secondary: 47A08 #47B37 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2012.11115
openalex publication_date 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a commuting d- tuple of operators \boldsymbol T defined on a complex separable Hilbert space \mathcal H, let [ [ \boldsymbol T^*, \boldsymbol T ] ] be the d× d block operator ( ( [ Tj^* , Ti ] ) ) of the commutators [T^*j , Ti] := T^*j Ti - TiTj^*. We define the determinant of [ [ \boldsymbol T^*, \boldsymbol T ] ] by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of [ [ \boldsymbol T^*, \boldsymbol 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 [ [ \boldsymbol T^*, \boldsymbol T ] ] must be 0. We show that if the d- tuple \boldsymbol T is cyclic, the determinant of [ [ \boldsymbol T^*, \boldsymbol 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 \mathcal H -- does not grow very rapidly, then the trace of the determinant of the operator [ [ \boldsymbol T^* , \boldsymbol 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.