Linear Operators: Spectral theory |
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Page 1378
... Theorem 23 is unique , and Pij - Pij , i , j = 1 , ... , k ; pi ; = 0 , if i > k or j > k . Pii PROOF . Suppose that σ , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ; } , i , j 1 ...
... Theorem 23 is unique , and Pij - Pij , i , j = 1 , ... , k ; pi ; = 0 , if i > k or j > k . Pii PROOF . Suppose that σ , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ; } , i , j 1 ...
Page 1379
... THEOREM . Let t , T , 4 , 01 , ... , on , etc. , be as in Theorem 18 . Then if , for j > k , the functions 0 † ( 2 ) of Theorem 18 ( or , the functions 05 ( 2 ) of Theorem 18 ) may be extended to analytic functions defined on the whole ...
... THEOREM . Let t , T , 4 , 01 , ... , on , etc. , be as in Theorem 18 . Then if , for j > k , the functions 0 † ( 2 ) of Theorem 18 ( or , the functions 05 ( 2 ) of Theorem 18 ) may be extended to analytic functions defined on the whole ...
Page 1904
... theorems , .IV.15 Alexandroff theorem on conver- gence of measures , ( 316 ) IV.9.15 Arzelą theorem on continuous limits , IV.6.11 ( 268 ) Banach theorem for operators into space of measurable functions , IV.11.2–3 ( 332–333 ) Egoroff ...
... theorems , .IV.15 Alexandroff theorem on conver- gence of measures , ( 316 ) IV.9.15 Arzelą theorem on continuous limits , IV.6.11 ( 268 ) Banach theorem for operators into space of measurable functions , IV.11.2–3 ( 332–333 ) Egoroff ...
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