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 ... COMPACT RESOLVENTS Spectral Theory: Compact Resolvents 1278.
... 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 ... COMPACT RESOLVENTS Spectral Theory: Compact Resolvents 1278.
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero