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 A , is the isometric automorphism in which maps [ x , y ] into [ y , x ] then П ( T − 1 ) = Â ̧Ã ( 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 A , is the isometric automorphism in which maps [ x , y ] into [ y , x ] then П ( T − 1 ) = Â ̧Ã ( T ) which shows that T is ...
Page 1393
... closed operator in Hilbert space . Then the set of complex numbers & such that the range of I - T is not closed is called the essential spectrum of T and is denoted by σ , ( T ) . It is clear that o , ( T ) Co ( T ) . If 7 is a formal ...
... closed operator in Hilbert space . Then the set of complex numbers & such that the range of I - T is not closed is called the essential spectrum of T and is denoted by σ , ( T ) . It is clear that o , ( T ) Co ( T ) . If 7 is a formal ...
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 ...
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