A Hilbert space problem book
Written for the active reader with some background in the topic, this book presents problems in Hilbert space theory, with definitions, corollaries and historical remarks, hints, proofs, answers and constructions.
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Vectors and Spaces
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adjoint algebra analytic functions approximate point spectrum assertion bilateral bounded linear transformation closed unit closure commutator compact operator complex numbers continuous convergence convex Corollary corresponding defined diagonal operator dilation dimension direct sum easy eigenvalues example finite finite-dimensional spaces follows functional Hilbert space Halmos hence Hilbert space hyponormal implies induced inequality infinite infinite-dimensional Hilbert space initial space inner product invariant subspaces invertible operator kernel linear functional linear transformation mapping multiplication operator necessary and sufficient nilpotent non-zero norm normal operator numerical range operator on H orthogonal complement orthonormal basis partial isometry polynomial positive integer positive number Problem proof is complete prove quadratic form quasinilpotent result scalar sequence Solution space H span spectral radius spectral theorem subnormal operator subset sufficient condition Suppose theory Toeplitz operators trivial uniform boundedness unilateral shift unit ball unit vector unitarily equivalent unitary operator vanishes vector space Volterra operator weak topology weighted shift