Linear Operators: Spectral operators |
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Page 898
... Borel set of complex numbers , then E ( 8 ) T = TE ( 8 ) , o ( Ts ) , where Ts is the restriction of T to E ( 8 ) ... bounded normal operator T and for each complex bounded Borel function on the spectrum o ( T ) let ( i ) f ( T ) = √。r ...
... Borel set of complex numbers , then E ( 8 ) T = TE ( 8 ) , o ( Ts ) , where Ts is the restriction of T to E ( 8 ) ... bounded normal operator T and for each complex bounded Borel function on the spectrum o ( T ) let ( i ) f ( T ) = √。r ...
Page 922
... Borel function defined on the complex plane we have f ( T2 ) → f ... bounded Borel functions on D. Let B be the B * -algebra , with norm [ f ] = SUPAED f ( 2 ) , of all complex bounded 922 X. BOUNDED NORMAL OPERATORS IN HILBERT SPACE X.7.1.
... Borel function defined on the complex plane we have f ( T2 ) → f ... bounded Borel functions on D. Let B be the B * -algebra , with norm [ f ] = SUPAED f ( 2 ) , of all complex bounded 922 X. BOUNDED NORMAL OPERATORS IN HILBERT SPACE X.7.1.
Page 1214
... bounded Borel set e of the real axis . Thus , if the function f in vanishes outside S , the quantity ( Vaf ) ( 2 ) = √ ̧ ̧ f ( s ) W ̧ ( s , 2 ) v ( ds ) satisfies the inequalities S。| ( V_ƒ ) ( 2 ) \ 2 , μ . ( d2 ) ≤ \ f \ 2 S ...
... bounded Borel set e of the real axis . Thus , if the function f in vanishes outside S , the quantity ( Vaf ) ( 2 ) = √ ̧ ̧ f ( s ) W ̧ ( s , 2 ) v ( ds ) satisfies the inequalities S。| ( V_ƒ ) ( 2 ) \ 2 , μ . ( d2 ) ≤ \ f \ 2 S ...
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 linear 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero