Linear Operators: Spectral theory |
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Page 861
... inverse map 7-1 is continuous . To see that t is also continuous it will first be shown that 7 ( X ) is closed in B ( X ) . To do this the following criterion is useful : an element Te B ( X ) is in 7 ( X ) if and only if ( Ty ) z = T ...
... inverse map 7-1 is continuous . To see that t is also continuous it will first be shown that 7 ( X ) is closed in B ( X ) . To do this the following criterion is useful : an element Te B ( X ) is in 7 ( X ) if and only if ( Ty ) z = T ...
Page 877
... inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. PROOF . If y1 exists as an element of Y then , since X and Y have the same unit , y ...
... inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. PROOF . If y1 exists as an element of Y then , since X and Y have the same unit , y ...
Page 1311
... inverse . This convenient assumption is equivalent to the supposition that the operator 7 has been replaced by τ - λ . Our first result is concerned with the number k of linearly in- dependent boundary conditions which define T. Notice ...
... inverse . This convenient assumption is equivalent to the supposition that the operator 7 has been replaced by τ - λ . Our first result is concerned with the number k of linearly in- dependent boundary conditions which define T. Notice ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero