A Hilbert Space Problem BookFrom the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem.... This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book." |
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adjoint algebra assertion bilateral bounded linear transformation closed unit closure commutator compact operator complex numbers continuous convergence convex Corollary defined diagonal operator dilation dimension direct sum easy eigenvalues example exists finite finite-dimensional spaces follows functional Hilbert space ƒ ² ƒ and g ƒ in H Halmos hence Hilbert space hyponormal implies inequality infinite infinite-dimensional Hilbert space initial space inner product invariant subspaces invertible operator kernel linear transformation mapping matrix measurable function 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 prove quadratic form result scalar sequence Solution space H span spectral radius spectral theorem subnormal operator subset sufficient condition theory Toeplitz operators trivial unilateral shift unit ball unit vector unitarily equivalent unitary operator vanishes vector f weak topology weakly weighted shift