Linear Operators, Part 2 |
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Page 888
... union of two commuting projections A and B. We recall that these operators are defined by the equations A ^ B = AB , A v B = A + B - AB and that the intersection and union of two commuting projections are again projection operators ...
... union of two commuting projections A and B. We recall that these operators are defined by the equations A ^ B = AB , A v B = A + B - AB and that the intersection and union of two commuting projections are again projection operators ...
Page 958
... union e is also in Bo . Let rn = en Uen + 10 . so that E ( r ) g0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g , y ( e , ) ) = dE ( e ) E ( r „ ) g → 0 . .... This argument shows that the vector valued additive set function y is ...
... union e is also in Bo . Let rn = en Uen + 10 . so that E ( r ) g0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g , y ( e , ) ) = dE ( e ) E ( r „ ) g → 0 . .... This argument shows that the vector valued additive set function y is ...
Page 1343
... union of a sequence of sets of finite μ - measure . By Lusin's lemma ( XII.3.17 ) each set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m ,, are continuous ...
... union of a sequence of sets of finite μ - measure . By Lusin's lemma ( XII.3.17 ) each set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m ,, are continuous ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero