Decomposition of the tensor product of two Hilbert modules

Abstract

Given a pair of positive real numbers α, β and a sesqui-analytic function K on a bounded domain Ω ⊂ C m , in this paper, we investigate the properties of the sesqui-analytic function K (α,β) , taking values in m × m matrices. One of the key findings is that K (α,β) is non-negative definite whenever K α and K β are non-negative definite. In this case, a realization of the Hilbert module determined by the kernel K (α,β) is obtained. Let Mi, i = 1, 2, be two Hilbert modules over the polynomial ring C[z1, . . . , zm]. Then C[z1, . . . , z2m] acts naturally on the tensor product M1 ⊗ M2. The restriction of this action to the polynomial ring C[z1, . . . , zm] obtained using the restriction map p → p |∆ leads to a natural decomposition of the tensor product M1 ⊗ M2, which is investigated. Two of the initial pieces in this decomposition are identified.

The irreducibility and strong irreducibility of operators

“…Proof. By Lemma 2.7 of [18], we know that if (M φ , H K 0 ) is bounded, then there exists c > 0 such that (c 2 − φ(z)φ(w))K 0 (z, w) is a nonnegative kernel on Ω × Ω. The minimum of all c satisfying this property is its norm.…”