Linear Operators, Part 2 |
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Page 1112
... formula and Cramer's formula for matrix inverses , we have d d d det ( A + zB ) | 2 = 0 = Σ Σ bij v ji dz = i = 1 j = 1 det ( 4 ) tr ( A - 1B ) , where Yii denotes the cofactor of the element a ;; of the matrix A. Substituting A = I + T ...
... formula and Cramer's formula for matrix inverses , we have d d d det ( A + zB ) | 2 = 0 = Σ Σ bij v ji dz = i = 1 j = 1 det ( 4 ) tr ( A - 1B ) , where Yii denotes the cofactor of the element a ;; of the matrix A. Substituting A = I + T ...
Page 1288
... formula was established for the case f , ge C ( I ) . However , the arguments are equally valid for f , ge H ( I ) . Q.E.D. It will be convenient for what follows to record other situations in which Green's formula is valid but where I ...
... formula was established for the case f , ge C ( I ) . However , the arguments are equally valid for f , ge H ( I ) . Q.E.D. It will be convenient for what follows to record other situations in which Green's formula is valid but where I ...
Page 1363
... formula 1 E ( ( 1 , 2 ) ) = lim lim Σπί 21 + 8 + 0-3 0-8 [ R ( 2 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
... formula 1 E ( ( 1 , 2 ) ) = lim lim Σπί 21 + 8 + 0-3 0-8 [ R ( 2 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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