Linear Operators: Spectral theory |
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Page 1020
... matrix obtained by deleting the ith row and the jth column in ( a ,, ) . Then det ( A ) = a ,, 4 , and 1a , Aik = 0 if jk . Assuming that A is one - to - one , Cramer's rule for 4-1 asserts that the matrix of det ( A ) A - 1 , relative ...
... matrix obtained by deleting the ith row and the jth column in ( a ,, ) . Then det ( A ) = a ,, 4 , and 1a , Aik = 0 if jk . Assuming that A is one - to - one , Cramer's rule for 4-1 asserts that the matrix of det ( A ) A - 1 , relative ...
Page 1080
... matrix elements of A are real . Let C = ( A - A * ) , and let g be the maximum of the absolute values of the matrix elements of C. Then - 192≤ g ' n ( n − 1 ) \ } 2 ( Hint : Use Exercise 33 and the case n = 2 of Exercise 31. ) 35 ...
... matrix elements of A are real . Let C = ( A - A * ) , and let g be the maximum of the absolute values of the matrix elements of C. Then - 192≤ g ' n ( n − 1 ) \ } 2 ( Hint : Use Exercise 33 and the case n = 2 of Exercise 31. ) 35 ...
Page 1338
... matrix measure whose elements μ are continuous with respect to a positive o - finite measure u . If the matrix of densities { m } is defined by the equations μ ,, ( e ) = f ̧m ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set , then ...
... matrix measure whose elements μ are continuous with respect to a positive o - finite measure u . If the matrix of densities { m } is defined by the equations μ ,, ( e ) = f ̧m ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set , then ...
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