Linear Operators: Spectral theory |
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Page 1187
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . PROOF . If A1 is the isometric automorphism in HH which maps [ x , y ] into [ y , x ] then П ( T - 1 ) = A , г ( T ) which shows that T is ...
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . PROOF . If A1 is the isometric automorphism in HH which maps [ x , y ] into [ y , x ] then П ( T - 1 ) = A , г ( T ) which shows that T is ...
Page 1436
... closed , has a closed range R. The inverse T1 of T1 is clearly closed , and is a one - to - one mapping of R into X. Thus by the closed graph theorem ( II.2.4 ) , T1 is bounded . It follows that if { f } is a sequence of elements of D ...
... closed , has a closed range R. The inverse T1 of T1 is clearly closed , and is a one - to - one mapping of R into X. Thus by the closed graph theorem ( II.2.4 ) , T1 is bounded . It follows that if { f } is a sequence of elements of D ...
Page 1902
... closed operators , VII.9.4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613–614 ) , ( 639–641 ) ... Closed curve , positive orientation of , ( 225 ) Closed graph theorem , II.2.4 ( 57 ) remarks on , ( 83-85 ) Closed linear ...
... closed operators , VII.9.4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613–614 ) , ( 639–641 ) ... Closed curve , positive orientation of , ( 225 ) Closed graph theorem , II.2.4 ( 57 ) remarks on , ( 83-85 ) Closed linear ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero