Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 80
Page 912
... Clearly f is 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 = Σ xa with a , a ,, and let U1 be ...
... Clearly f is 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 = Σ xa with a , a ,, and let U1 be ...
Page 1252
... Clearly B is a normed linear space under this norm . Let be the closure in B ** of κ ( B ) , where x is the natural isometric imbedding of B in B ** ( cf. II.3.19 ) . It is seen from Lemma 1.6.7 that is complete . We shall show is a ...
... Clearly B is a normed linear space under this norm . Let be the closure in B ** of κ ( B ) , where x is the natural isometric imbedding of B in B ** ( cf. II.3.19 ) . It is seen from Lemma 1.6.7 that is complete . We shall show is a ...
Page 1428
... clearly C = sup [ ≤ f ( t ) 2dt sup 0≤m < ∞∞ 0≤m < ∞ 1 m - 1 if ( t ) | 2dt + 2M2 < \ f \ 2 + 2M2 < ∞ . By the preceding lemma we have | f ( k ) | = m 0 ( 1m ) / " ) [ * ] and [ ** ] \ † m ( k ) \ 2 \\ ƒm11 ) \ 2 ≤ C1 / 2 − k ...
... clearly C = sup [ ≤ f ( t ) 2dt sup 0≤m < ∞∞ 0≤m < ∞ 1 m - 1 if ( t ) | 2dt + 2M2 < \ f \ 2 + 2M2 < ∞ . By the preceding lemma we have | f ( k ) | = m 0 ( 1m ) / " ) [ * ] and [ ** ] \ † m ( k ) \ 2 \\ ƒm11 ) \ 2 ≤ C1 / 2 − k ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
52 other sections not shown
Other editions - View all
Common terms and phrases
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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero