Cyclic Operators, Linear Functionals and RKHS

2025/07/23 by Wang, Yi
#47B02 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.17358

Abstract

Given a commuting n-tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional Λ_T,h on the polynomial ring ℂ[z,z]. ``Near subnormality properties'' of an operator T are translated into positivity properties of Λ<sub>T,h</sub>. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is Λ_T,h given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function F_T,h on ℂⁿ. Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function F_T,h, we give a function model for all cyclic commuting n-tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of F_T,h, but one may also replace the Fock space by other RKHS. We give many examples in the last section.

Citations

• An invitation to the Drury-Arveson space
• Decomposing algebraic m-isometric tuples
• An Analytic Grothendieck Riemann Roch Theorem
• Geometric Arveson-Douglas conjecture
• Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
• A new kind of index theorem
• The Dirac operator of a commuting d-tuple
• Subalgebras of C*-algebras III: multivariable operator theory

Related

• Cyclicity in reproducing kernel Hilbert spaces of analytic functions
• Four Properties of Reproducing Kernel Hilbert Spaces
• A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces
• Operator Positivity and Analytic Models of Commuting Tuples of Operators
• Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations