Linear Operators, Part 2 |
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Page 906
... positive definite if it is positive and ( Tx , x ) > 0 for every x 0 in H. - = It is clear that all of these classes of operators are normal . Unitary operators have a number of other characteristic proper- ties . For example , if U is ...
... positive definite if it is positive and ( Tx , x ) > 0 for every x 0 in H. - = It is clear that all of these classes of operators are normal . Unitary operators have a number of other characteristic proper- ties . For example , if U is ...
Page 1247
... positive self adjoint transformations and their square roots . 2 LEMMA . A self adjoint transformation T is positive if and only if o ( T ) is a subset of the interval [ 0 , ∞ ) . PROOF . Let E be the resolution of the identity for T ...
... positive self adjoint transformations and their square roots . 2 LEMMA . A self adjoint transformation T is positive if and only if o ( T ) is a subset of the interval [ 0 , ∞ ) . PROOF . Let E be the resolution of the identity for T ...
Page 1338
... positive matrix measure whose elements μ , are continuous with respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ ( e ) = [ m ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set ...
... positive matrix measure whose elements μ , are continuous with respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ ( e ) = [ m ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero