Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 905
... orthonormal basis B for eigenvectors of T. Furthermore , consisting of x = Σ ( x , y ) y , ve B xЄ H , and , for ... orthonormal basis B. If we let B = n 1 B2 , then every element of B is an eigenvector of T. Since E ( u ) xx for x in B ...
... orthonormal basis B for eigenvectors of T. Furthermore , consisting of x = Σ ( x , y ) y , ve B xЄ H , and , for ... orthonormal basis B. If we let B = n 1 B2 , then every element of B is an eigenvector of T. Since E ( u ) xx for x in B ...
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 . 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 . LEMMA . The Hilbert - Schmidt norm is independent of the orthonormal basis used in its definition ...
Page 1028
... orthonormal basis for H. Since E§ 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 e A - B } is an orthonormal ...
... orthonormal basis for H. Since E§ 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 e A - B } is an orthonormal ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero