Linear Operators: Spectral theory |
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Page 1187
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . n n PROOF . If A is the isometric automorphism in which maps [ x , y ] into [ y , x ] then ( T - 1 ) = A , T ( T ) which shows that T is ...
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . n n PROOF . If A is the isometric automorphism in which maps [ x , y ] into [ y , x ] then ( T - 1 ) = A , T ( T ) which shows that T is ...
Page 1226
... closed symmetric extension of a symmetric operator T with dense domain is called its closure , and written T. 8 ... closed extensions . ( c ) The operator T and its closure have the same adjoint . ( d ) An operator is closed if and only ...
... closed symmetric extension of a symmetric operator T with dense domain is called its closure , and written T. 8 ... closed extensions . ( c ) The operator T and its closure have the same adjoint . ( d ) An operator is closed if and only ...
Page 1393
... closed operator in Hilbert space . Then the set of complex numbers λ such that the range of I - T is not closed is called the essential spectrum of T and is denoted by σ 。( T ) . It is clear that o , ( T ) Co ( T ) . If t is a formal ...
... closed operator in Hilbert space . Then the set of complex numbers λ such that the range of I - T is not closed is called the essential spectrum of T and is denoted by σ 。( T ) . It is clear that o , ( T ) Co ( T ) . If t is a formal ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero