Linear Operators: Spectral theory |
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Page 1010
... orthonormal sequence was used . The following lemma shows that the class HS depends only upon the Hilbert space and not upon the basis . 2 LEMMA . The Hilbert - Schmidt norm is independent of the orthonormal basis used in its definition ...
... orthonormal sequence was used . The following lemma shows that the class HS depends only upon the Hilbert space and not upon the basis . 2 LEMMA . The Hilbert - Schmidt norm is independent of the orthonormal basis used in its definition ...
Page 1028
... orthonormal basis for H. Since ES is finite dimensional we may suppose without loss of generality that there is a finite subset B of A such that { x , α = B } is an orthonormal basis for E§ , and { x , α € A — B } is an orthonormal ...
... orthonormal basis for H. Since ES is finite dimensional we may suppose without loss of generality that there is a finite subset B of A such that { x , α = B } is an orthonormal basis for E§ , and { x , α € A — B } is an orthonormal ...
Page 1779
... orthonormal basis for the linear manifold N in H if A is an orthonormal set contained in N and if x Є N. x = Σ ( x , y ) y , VEA 12 THEOREM . Every closed linear manifold in § contains an orthonormal basis for itself . PROOF . If the ...
... orthonormal basis for the linear manifold N in H if A is an orthonormal set contained in N and if x Є N. x = Σ ( x , y ) y , VEA 12 THEOREM . Every closed linear manifold in § contains an orthonormal basis for itself . PROOF . If the ...
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