Linear Operators, Part 2 |
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Page 1037
... uniform convergence of { ( T ) } on C and the maximum modulus principle imply that the convergence is uniform inside C , even though C may enclose points of o ( T ) . Hence ( Tn ) → ( T ) for all λ 0 , proving ‡ that the mapping T → λ ...
... uniform convergence of { ( T ) } on C and the maximum modulus principle imply that the convergence is uniform inside C , even though C may enclose points of o ( T ) . Hence ( Tn ) → ( T ) for all λ 0 , proving ‡ that the mapping T → λ ...
Page 1095
... uniform topology of operators , ( Corollary VI.5.5 ) , there exists a compact operator T such that T → T in the uniform topology . Thus , by Corollary 4 ( a ) , limm → ∞μx ( Tn — Tm ) = μ¿ ( Tn — T ) . It follows that = N Σμ . ( Τ ...
... uniform topology of operators , ( Corollary VI.5.5 ) , there exists a compact operator T such that T → T in the uniform topology . Thus , by Corollary 4 ( a ) , limm → ∞μx ( Tn — Tm ) = μ¿ ( Tn — T ) . It follows that = N Σμ . ( Τ ...
Page 1922
... Uniform boundedness principle , in B- spaces , II.3.20-21 ( 66 ) discussion of , ( 80–82 ) in F - spaces , II.1.11 ( 52 ) for measures , IV.9.8 ( 309 ) Uniform continuity , of an almost pe- riodic function , IV.7.4 ( 283 ) criterion for ...
... Uniform boundedness principle , in B- spaces , II.3.20-21 ( 66 ) discussion of , ( 80–82 ) in F - spaces , II.1.11 ( 52 ) for measures , IV.9.8 ( 309 ) Uniform continuity , of an almost pe- riodic function , IV.7.4 ( 283 ) criterion for ...
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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