Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 91
Page 1020
... matrix obtained by deleting the ith row and the jth column in ( a ,, ) . Then det ( 4 ) = Σ_a1 , A1 , and Σ - 1a¡¡ Aik = 0 if j ‡ k . Assuming that A is one - to - one , Cramer's rule for A - 1 asserts that the matrix of det ( A ) A - 1 ...
... matrix obtained by deleting the ith row and the jth column in ( a ,, ) . Then det ( 4 ) = Σ_a1 , A1 , and Σ - 1a¡¡ Aik = 0 if j ‡ k . Assuming that A is one - to - one , Cramer's rule for A - 1 asserts that the matrix of det ( A ) A - 1 ...
Page 1338
... matrix measure whose elements μ are continuous with respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ ,, ( e ) = [ _m ,, ( ^ ) μ ( dλ ) , where e is any bounded Borel set , then ...
... matrix measure whose elements μ are continuous with respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ ,, ( e ) = [ _m ,, ( ^ ) μ ( dλ ) , where e is any bounded Borel set , then ...
Page 1378
... matrix measure { p .; } which by Corollary 21 must be the same as the matrix measure of Theorem 13. In particular , ô , is unique . Thus , all that remains for us to prove is that if p ,, ( e ) = 0 for jk , then σ1 , ... , σ is a ...
... matrix measure { p .; } which by Corollary 21 must be the same as the matrix measure of Theorem 13. In particular , ô , is unique . Thus , all that remains for us to prove is that if p ,, ( e ) = 0 for jk , then σ1 , ... , σ is a ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
57 other sections not shown
Other editions - View all
Common terms and phrases
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