Linear Operators: Spectral theory |
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Page 879
... operators in . This leads us to the notion of a normal operator as given in the following definition . 14 DEFINITION . A bounded linear operator T in a Hilbert space is said to be normal if TT * = T * T , and self adjoint if T = T ...
... operators in . This leads us to the notion of a normal operator as given in the following definition . 14 DEFINITION . A bounded linear operator T in a Hilbert space is said to be normal if TT * = T * T , and self adjoint if T = T ...
Page 930
... operator , distinct from the zero and identity operators , has a non - trivial invariant subspace . It is readily seen from Theorem VII.3.10 that if T is a bounded linear operator in a B - space X and if o ( T ) contains at least two ...
... operator , distinct from the zero and identity operators , has a non - trivial invariant subspace . It is readily seen from Theorem VII.3.10 that if T is a bounded linear operator in a B - space X and if o ( T ) contains at least two ...
Page 935
... operators to be stated here . An operator U in a Hilbert space is called a partial isometry with initial domain M if | Ux | = | x | for x EM and Ux = 0 for x = H↔ M. THEOREM . Every bounded linear transformation T in a Hilbert space ...
... operators to be stated here . An operator U in a Hilbert space is called a partial isometry with initial domain M if | Ux | = | x | for x EM and Ux = 0 for x = H↔ M. THEOREM . Every bounded linear transformation T in a Hilbert space ...
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