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 ( 4 ) = a ,, 4 , and Σa ;; Ak = 0 if jk . 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 ) = a ,, 4 , and Σa ;; Ak = 0 if jk . Assuming that A is one - to - one , Cramer's rule for A - 1 asserts that the matrix of det ( A ) A - 1 ...
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 92 ≤ g ( n ( n − 1 ) ` 2 ( Hint : Use Exercise 33 and the case n = 2 of Exercise 31. ) 35 ( Pick ) ...
... 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 92 ≤ g ( n ( n − 1 ) ` 2 ( Hint : Use Exercise 33 and the case n = 2 of Exercise 31. ) 35 ( Pick ) ...
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 M1 , ( e ) = √ m1 , ( 2 ) μ ( d2 ) , where e is any bounded Borel set ...
... 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 M1 , ( e ) = √ m1 , ( 2 ) μ ( d2 ) , where e is any bounded Borel set ...
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