Holomorphic kernels and commuting operators

Athavale, Ameer

doi:10.1090/S0002-9947-1987-0906808-6

Holomorphic kernels and commuting operators
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by Ameer Athavale PDF

Trans. Amer. Math. Soc. 304 (1987), 101-110 Request permission

Abstract:

Necessary and sufficient conditions in terms of operator polynomials are obtained for an $m$-tuple $T = ({T_1}, \ldots ,{T_m})$ of commuting bounded linear operators on a separable Hilbert space $\mathcal {H}$ to extend to an $\dot m$-tuple $S = ({S_1}, \ldots ,{S_m})$ of operators on some Hilbert space $\mathcal {K}$, where each ${S_i}$ is realized as a ${\ast }$-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for $T$ to have a commuting normal extension.

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