Linear Operators: Spectral theory |
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Page 905
... orthonormal basis B. If we let B n Un - 1 Bn then every element of B is an eigenvector of T. Since E ( u ) x B , and E ( un ) E ( μm ) = x for x in 0 if nm , we see that B is an orthonormal set . Also , by Corollary 2.3 , x = ∞ ΣΕ ( μη ) ...
... orthonormal basis B. If we let B n Un - 1 Bn then every element of B is an eigenvector of T. Since E ( u ) x B , and E ( un ) E ( μm ) = x for x in 0 if nm , we see that B is an orthonormal set . Also , by Corollary 2.3 , x = ∞ ΣΕ ( μη ) ...
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 E is finite dimensional we may suppose without loss of generality that there is a finite subset B of A such that { ~ , α = B } is an orthonormal basis for ES , and { x , a Є 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 { ~ , α = B } is an orthonormal basis for ES , and { x , a Є A − B } is an orthonormal ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero