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Vectors and Spaces
Real functions in
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adjoint algebraic assertion assumption bilateral closure commutator compact operator complex numbers Conclusion continuous convergence convex Corollary corresponding countable cyclic vector defined dense diagonal operator dimension direct sum easy eigenvalues equal everywhere example exists finite-dimensional spaces follows functional Hilbert space Halmos hence Hilbert space implies induced inequality infinite infinite-dimensional Hilbert space initial space inner product integral operator invariant subspaces invertible operator kernel linear functional linear transformation mapping necessary and sufficient nilpotent non-trivial non-zero norm normal extension normal operator numerical range operator on H orthogonal complement orthonormal basis partial isometry polynomial positive integer positive number Problem proof is complete quadratic form quasinilpotent restriction 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