Linear Operators: Spectral theory |
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Page 932
... Extension of operators . Let A be an operator in a Hilbert space H. It is of interest to obtain conditions under which A has a normal extension to some Hilbert space 2 , i.e. , when there exists a normal operator B on & such that if P ...
... Extension of operators . Let A be an operator in a Hilbert space H. It is of interest to obtain conditions under which A has a normal extension to some Hilbert space 2 , i.e. , when there exists a normal operator B on & such that if P ...
Page 1239
... extension of T. Then by Lemma 26 , T1 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 ...
... extension of T. Then by Lemma 26 , T1 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 ...
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 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero