Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 ...
... 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 ...
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 ) = fm ,, ( 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 ) = fm ,, ( 2 ) μ ( d2 ) , where e is any bounded Borel set ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 Co(I coefficients compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element essential spectrum 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(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero