Linear Operators, Part 2 |
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Page 925
... imply ( c ) A1 A2 ≤ and B1 ≤ B2 imply A + B1 ≤ A2 + B2 ; ( d ) A≤ B and imply aA ≤ aB ; ( e ) A≤ B implies -B≤ - A ; ( f ) if A is Hermitian , there are numbers m and M such that mI ≤ A≤MI . Show that 0 ≤ A≤ B implies that 42 ...
... imply ( c ) A1 A2 ≤ and B1 ≤ B2 imply A + B1 ≤ A2 + B2 ; ( d ) A≤ B and imply aA ≤ aB ; ( e ) A≤ B implies -B≤ - A ; ( f ) if A is Hermitian , there are numbers m and M such that mI ≤ A≤MI . Show that 0 ≤ A≤ B implies that 42 ...
Page 1124
... implies E = E1 . - > Ex , and E → ∞ n ∞ n Similarly , q ( E ) ≤ q ( E1 ) implies E ≤ E1 . If E „ , 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 ...
... implies E = E1 . - > Ex , and E → ∞ n ∞ n Similarly , q ( E ) ≤ q ( E1 ) implies E ≤ E1 . If E „ , 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 ...
Page 1229
... implies that d2d 2 for all s = d + d_ e isometric mapping of a subspace 1 ; i.e. , that 1 is the graph of an of D onto a subspace of D_ . + On the other hand , if | d | 2 | d_2 for s = d ++ d_e1 , we have \ d ++ e_ | 2 = \ d_ + e_ | 2 ...
... implies that d2d 2 for all s = d + d_ e isometric mapping of a subspace 1 ; i.e. , that 1 is the graph of an of D onto a subspace of D_ . + On the other hand , if | d | 2 | d_2 for s = d ++ d_e1 , we have \ d ++ e_ | 2 = \ d_ + e_ | 2 ...
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