Linear Operators: Spectral theory |
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Page 942
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = μ ( E ) it is seen that 9 √q8 ( su ̄1 ) q ( ut ) μ ( du ) = 2q ( st ) , i.e. , every translate q ' of an eigenfunction corresponding to λ is also ...
... continuous function . By replacing s by st and u by ut and using the fact that μ ( Et ) = μ ( E ) it is seen that 9 √q8 ( su ̄1 ) q ( ut ) μ ( du ) = 2q ( st ) , i.e. , every translate q ' of an eigenfunction corresponding to λ is also ...
Page 1002
... function may be approximated uniformly by linear combinations of functions of the form exp i ( t , x1 + t2x2 ) . 6 A continuous function f on a topological group G is called almost periodic if for each ɛ > 0 there exists a compact set K ...
... function may be approximated uniformly by linear combinations of functions of the form exp i ( t , x1 + t2x2 ) . 6 A continuous function f on a topological group G is called almost periodic if for each ɛ > 0 there exists a compact set K ...
Page 1903
... Continuous functions . ( See also Abso- lutely continuous functions ) as a B - space , additional properties , IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6 characterizations of C - space , ( 396 ...
... Continuous functions . ( See also Abso- lutely continuous functions ) as a B - space , additional properties , IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6 characterizations of C - space , ( 396 ...
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