Linear Operators: Spectral operators |
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Page 929
... subspaces of X with respect to T , then M is said to reduce T. It is not difficult to see that a non - trivial subspace of Hilbert space may be an invariant subspace for an opera- tor but not reduce the operator . In fact , an operator ...
... subspaces of X with respect to T , then M is said to reduce T. It is not difficult to see that a non - trivial subspace of Hilbert space may be an invariant subspace for an opera- tor but not reduce the operator . In fact , an operator ...
Page 930
... subspaces for a given operator . It is not known whether every operator , distinct from the zero and identity operators , has a non - trivial invariant subspace . It is readily seen from Theorem VII.3.10 that if T is a bounded linear ...
... subspaces for a given operator . It is not known whether every operator , distinct from the zero and identity operators , has a non - trivial invariant subspace . It is readily seen from Theorem VII.3.10 that if T is a bounded linear ...
Page 1228
... subspace of D ( T * ) in- cluding D ( T ) , put S1 = S ( D , OD_ ) . Clearly , 1 is closed and symmetric , and Q D ... subspace of D + D , and D ( T ) → S1 . ( a ) The space is symmetric if and only if S1 is the graph of an isometric ...
... subspace of D ( T * ) in- cluding D ( T ) , put S1 = S ( D , OD_ ) . Clearly , 1 is closed and symmetric , and Q D ... subspace of D + D , and D ( T ) → S1 . ( a ) The space is symmetric if and only if S1 is the graph of an isometric ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero