A Hilbert space problem book
From 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 assumption bilateral bounded linear transformation closure commutes compact operator complex numbers Conclusion continuous convergence convex Corollary countable cyclic vector defined dense diagonal operator dilation dimension direct sum easy eigenvalues equal everywhere example exists finite-dimensional spaces follows functional Hilbert space Halmos hence Hilbert space hyponormal implies induced inequality infinite infinite-dimensional Hilbert space initial space inner product integral operator invariant subspaces invertible operators kernel linear transformation mapping multiplication operator necessary and sufficient nilpotent non-trivial 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 quasinilpotent result scalar sequence Solution space H span spectral radius spectral theorem strongly subnormal operator subset sufficient condition Suppose theory Toeplitz operators trivial unilateral shift unit ball unit circle unit vector unitarily equivalent unitary operator vanishes Volterra operator weak topology weakly weighted shift