Linear Operators: Spectral theory |
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Page 1378
... Theorem 23 is unique , and Pij == = Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . - Pii Ok PROOF . Suppose that σ1 , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { pu } , i , j = 1 ...
... Theorem 23 is unique , and Pij == = Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . - Pii Ok PROOF . Suppose that σ1 , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { pu } , i , j = 1 ...
Page 1379
... THEOREM . Let τ , T , A , σ1 , . on , etc. , be as in Theorem 18 . Then if , for j > k , the functions ( 2 ) of Theorem 18 ( or , the functions 05 ( 2 ) of Theorem 18 ) may be extended to analytic functions defined on the whole ...
... THEOREM . Let τ , T , A , σ1 , . on , etc. , be as in Theorem 18 . Then if , for j > k , the functions ( 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- IV.9.15 gence of measures , ( 316 ) Arzelà 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- IV.9.15 gence of measures , ( 316 ) Arzelà 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
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 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 open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero