Linear Operators: Spectral theory |
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Page 1024
... seen that ( I — B ̧ ) −1 [ ( x , y ) = [ ( 1 − B ) -1 a , ( I – B ) −1æ , 1 -1 — [ ( IB ) -1x , ( 1 + + tr ( B ) ) TM y ] . Therefore ( IB ) -1 | ≤ | ( I — Bx ) −1 and so N ( ii ) | det ( I — By ) || ( I — B ) −1 | ≤ | det ...
... seen that ( I — B ̧ ) −1 [ ( x , y ) = [ ( 1 − B ) -1 a , ( I – B ) −1æ , 1 -1 — [ ( IB ) -1x , ( 1 + + tr ( B ) ) TM y ] . Therefore ( IB ) -1 | ≤ | ( I — Bx ) −1 and so N ( ii ) | det ( I — By ) || ( I — B ) −1 | ≤ | det ...
Page 1154
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is the homeomorphic homomorphism in R ( 2 ) ...
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is the homeomorphic homomorphism in R ( 2 ) ...
Page 1324
... seen ( cf. Theorem 10 ) that Ta = 0 , i = 1 , . . n . Thus choosing a basis { } , i 1 . .... n , for the solutions of tσ = 0 , and defining the matrix { T } by the equations ai n i = 1 , ... , n , j = 1 the jump equations are seen to be ...
... seen ( cf. Theorem 10 ) that Ta = 0 , i = 1 , . . n . Thus choosing a basis { } , i 1 . .... n , for the solutions of tσ = 0 , and defining the matrix { T } by the equations ai n i = 1 , ... , n , j = 1 the jump equations are seen to be ...
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