Linear Operators: Spectral operators |
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Page 1239
... extension of T. Then by Lemma 26 , T , is the restriction of T * to a subspace W of D ( T * ) determined by a symmetric family of linearly independent boundary conditions B ( x ) = 0 , i 1 , ... , k , and we have only to show that k = n ...
... extension of T. Then by Lemma 26 , T , is the restriction of T * to a subspace W of D ( T * ) determined by a symmetric family of linearly independent boundary conditions B ( x ) = 0 , i 1 , ... , k , and we have only to show that k = n ...
Page 1270
... extension is to search for self adjoint extensions but to allow the extended operator to act in a Hilbert space containing the original one . In Section X.9 we discussed some related problems , considered by Naimark [ 3 ] , Sz . - Nagy ...
... extension is to search for self adjoint extensions but to allow the extended operator to act in a Hilbert space containing the original one . In Section X.9 we discussed some related problems , considered by Naimark [ 3 ] , Sz . - Nagy ...
Page 1397
... extension T of To ( t ) is independent of the particular extension chosen , i.e. , is independent of the particular set of boundary conditions defining this extension . We shall now show that the isolated points of o ( T ) depend quite ...
... extension T of To ( t ) is independent of the particular extension chosen , i.e. , is independent of the particular set of boundary conditions defining this extension . We shall now show that the isolated points of o ( T ) depend quite ...
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