Linear Operators, Part 2 |
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Page 1020
... matrix obtained by deleting the ith row and the jth column in ( a ,, ) . Then det ( 4 ) = 1 a ,, 4 , and Σa1 ; 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 ) = 1 a ,, 4 , and Σa1 ; 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 $ 2≤ g ' n ( n⋅ 2 - 1 ) \ ( 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 $ 2≤ g ' n ( n⋅ 2 - 1 ) \ ( 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 μ ( e ) = [ m ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set , then the ...
... 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 ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set , then the ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero