Linear Operators: Spectral theory |
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Page 940
... G -Solof ( str ( ds ) ) u ( dt ) = = [ f ( st ) , ( ds ) = f ( s ) v , ( ds ) , Sqf ( st ) v1 ( ds ) = √qt ( s ) v1 ... field of scalars is taken to be the field of complex numbers . 4 THEOREM . ( Peter - Weyl ) Let G be a compact ...
... G -Solof ( str ( ds ) ) u ( dt ) = = [ f ( st ) , ( ds ) = f ( s ) v , ( ds ) , Sqf ( st ) v1 ( ds ) = √qt ( s ) v1 ... field of scalars is taken to be the field of complex numbers . 4 THEOREM . ( Peter - Weyl ) Let G be a compact ...
Page 944
... g ( tu ) μ ( dt ) , .. .9 which means that ( Tg ) " = Th ( g " ) . Now if g “ 1 , gun form a basis for the space of ... field and μ its Haar measure . Then the set of continuous characters is fundamental both in C ( G ) and in L2 ( G , Σ ...
... g ( tu ) μ ( dt ) , .. .9 which means that ( Tg ) " = Th ( g " ) . Now if g “ 1 , gun form a basis for the space of ... field and μ its Haar measure . Then the set of continuous characters is fundamental both in C ( G ) and in L2 ( G , Σ ...
Page 1153
... field of Borel sets . Then if f is λ - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2x2 measurable . = PROOF . Let ExΣ be the product o - field of subsets of R R and let E be an open subset of R. Then , because of ...
... field of Borel sets . Then if f is λ - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2x2 measurable . = PROOF . Let ExΣ be the product o - field of subsets of R R and let E be an open subset of R. Then , because of ...
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