Linear Operators: Spectral theory |
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Page 911
... measure E defined by the equation E ( e ) = U − 1P ( e ) U is a resolution of the identity for T. = It is often convenient to express the result of Theorem 3 some- what differently . 4 COROLLARY . Let T be a normal operator in the ...
... measure E defined by the equation E ( e ) = U − 1P ( e ) U is a resolution of the identity for T. = It is often convenient to express the result of Theorem 3 some- what differently . 4 COROLLARY . Let T be a normal operator in the ...
Page 912
... u - measurable , and since the norm of the restriction of T to H is at most | T | , it follows from Lemma 3.2 that | f ( s ) | ≤ | T | . Let D1 consist of all finite sums of the form x = Σxas with ← H 。,, and let U1 be the map from ...
... u - measurable , and since the norm of the restriction of T to H is at most | T | , it follows from Lemma 3.2 that | f ( s ) | ≤ | T | . Let D1 consist of all finite sums of the form x = Σxas with ← H 。,, and let U1 be the map from ...
Page 1341
... measurable functions q1 , i = 1 , ... , n , μ - integrable over each bounded interval , and μ - measurable functions a¿¡ , 1 ≤ i , j ≤n , such that for u - almost all λ , and ( a ) ( b ) n n p¡ , Σ a ,, ( 2 ) a ,, ( 2 ) = dμ ‚ j = 1 ...
... measurable functions q1 , i = 1 , ... , n , μ - integrable over each bounded interval , and μ - measurable functions a¿¡ , 1 ≤ i , j ≤n , such that for u - almost all λ , and ( a ) ( b ) n n p¡ , Σ a ,, ( 2 ) a ,, ( 2 ) = dμ ‚ j = 1 ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 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 open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero