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Hankel operators and invariant subspaces of the Dirichlet space
Author(s) -
Luo Shuaibing,
Richter Stefan
Publication year - 2015
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdv001
Subject(s) - linear subspace , mathematics , square integrable function , invariant (physics) , pure mathematics , dirichlet l function , dirichlet's energy , invariant subspace , dirichlet kernel , reflexive operator algebra , dirichlet eigenvalue , dirichlet's principle , standard probability space , integrable system , dirichlet distribution , mathematical analysis , mathematical physics , computer science , programming language , compact operator , boundary value problem , extension (predicate logic)
The Dirichlet space D is the space of all analytic functions f on the open unit disc D such that f ' is square integrable with respect to two‐dimensional Lebesgue measure. In this paper, we prove that the invariant subspaces of the Dirichlet shift are in one‐to‐one correspondence with the kernels of the Dirichlet–Hankel operators. We then apply this result to obtain information about the invariant subspace lattice of the weak product D ⊙ D and to some questions about approximation of invariant subspaces of D . Our main results hold in the context of superharmonically weighted Dirichlet spaces.