Linear Operators: Spectral theory |
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Page 950
... compact Abelian group which we denote by R. We assume throughout that R is o - compact , i.e. , the union of countably many compact sets . Every such group has a non - negative countably additive measure which is defined on the Borel ...
... compact Abelian group which we denote by R. We assume throughout that R is o - compact , i.e. , the union of countably many compact sets . Every such group has a non - negative countably additive measure which is defined on the Borel ...
Page 1150
... compact , o - compact Abelian group . However , there are a few comments that we should make con- cerning the general non - Abelian case . First of all we shall prove , in Theorem 2 , that a locally compact group is automatically a ...
... compact , o - compact Abelian group . However , there are a few comments that we should make con- cerning the general non - Abelian case . First of all we shall prove , in Theorem 2 , that a locally compact group is automatically a ...
Page 1331
... compact . This is a special case of Exercise VI.9.52 , but , for the sake of completeness , a proof will be given here . Note first , that by Schwarz ' inequality , √ , \ √ , K ( t ... COMPACT RESOLVENTS Spectral Theory: Compact Resolvents.
... compact . This is a special case of Exercise VI.9.52 , but , for the sake of completeness , a proof will be given here . Note first , that by Schwarz ' inequality , √ , \ √ , K ( t ... COMPACT RESOLVENTS Spectral Theory: Compact Resolvents.
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