Linear Operators: Spectral theory |
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Page 925
... imply A≤ C ; ( c ) A , A , and B1 ≤ B2 imply A1 + B1 ≤ A2 + B2 ; ( d ) A≤ B and α 20 imply aA ≤aB ; ( e ) A≤ B implies - -B≤ − A ; ( f ) if A is Hermitian , there are numbers m and M such that ml ≤ A≤ MI . 18 Show that 0 ≤ A≤ B ...
... imply A≤ C ; ( c ) A , A , and B1 ≤ B2 imply A1 + B1 ≤ A2 + B2 ; ( d ) A≤ B and α 20 imply aA ≤aB ; ( e ) A≤ B implies - -B≤ − A ; ( f ) if A is Hermitian , there are numbers m and M such that ml ≤ A≤ MI . 18 Show that 0 ≤ A≤ B ...
Page 1124
... implies E = E. Similarly , q ( E ) ≤ q ( E1 ) implies E≤ E1 . If En , E are in F and ( E ) increases to the limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If ...
... implies E = E. Similarly , q ( E ) ≤ q ( E1 ) implies E≤ E1 . If En , E are in F and ( E ) increases to the limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If ...
Page 1402
... implies ( c ) . It is clear that ( c ) implies ( a ) . Thus the proof is complete . Q.E.D. The following corollary is contained in the preceding proof : 12 COROLLARY . If the deficiency indices of t are ( n , n ) , then the essential ...
... implies ( c ) . It is clear that ( c ) implies ( a ) . Thus the proof is complete . Q.E.D. The following corollary is contained in the preceding proof : 12 COROLLARY . If the deficiency indices of t are ( n , n ) , then the essential ...
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