Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 888
... projections are again projection operators . Also the ranges of the intersection and union of two commuting projection operators are given by the equations ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X ( 4X ) + ( BX ) = sp ( AX , BX ) ...
... projections are again projection operators . Also the ranges of the intersection and union of two commuting projection operators are given by the equations ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X ( 4X ) + ( BX ) = sp ( AX , BX ) ...
Page 1123
... projection for T if T leaves the range of E invariant , i.e. , if ETE = TE . 3 LEMMA . Any operator T in Hilbert ... projection E. If E , E1 are in F , 1 implies E = E1 . = ∞ = and ( XI.10.2 1123 SUBDIAGONALIZATION OF COMPACT OPERATORS.
... projection for T if T leaves the range of E invariant , i.e. , if ETE = TE . 3 LEMMA . Any operator T in Hilbert ... projection E. If E , E1 are in F , 1 implies E = E1 . = ∞ = and ( XI.10.2 1123 SUBDIAGONALIZATION OF COMPACT OPERATORS.
Page 1126
... projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections E ,, it follows from ( 1 ) that the closure in ( m ) of the vectors ( 4 ) is ( a ) . Thus ...
... projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections E ,, it follows from ( 1 ) that the closure in ( m ) of the vectors ( 4 ) is ( a ) . Thus ...
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