A Trace Inequality for Commuting d-Tuples of Operators

Abstract

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting d-tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a * -algebra representation ρ of C 0 (S) (where S is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on S and is adapted to the * -representation ρ via a unitary representation U of the group G, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation U becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space S ⊂ C d decomposes into finitely many G-orbits. In such cases, the imprimitivity based on S admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of G-orbits.

Operator theory on generalized Hartogs triangles

“…• finitely rationally cyclic if for some finite subset F of H, H = {r(T )h : r is a rational function with poles off σ(T ) and h ∈ F }, Following [18], we define the determinant operator det ([T * , T ]) of a commuting pair…”

“…The reader is referred to [2,11,18] for some basic facts pertaining to rational cyclicity, essential normality and related notions.…”

“…In this subsection, we obtain trace estimates of the determinant operator of the multiplication 2-tuple M z for a family of reproducing kernel Hilbert spaces on generalized Hartogs triangles (cf. [18,Definition 4.4]). The reader is referred to [26] for the basic properties of trace class operators and related notions.…”

Trace Formula of Semicommutators

“…Various properties, such as compactness, Schatten class membership, trace formulas, were studied in a numerous amount of works (c.f. [1,7,11,12,20,25,31,33,38,39]). Among others, it is well-known (c.f.…”