Linear Operators: Spectral theory |
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Page 890
... of the spectrum { 21 , . 2 } of T and otherwise let E ( 8 ) be the sum of all the projections E ( 2 , ) for which 2 € 8 , then the function E is a resolution of the identity for T and the operational calculus is given by the formula ( vi ) ...
... of the spectrum { 21 , . 2 } of T and otherwise let E ( 8 ) be the sum of all the projections E ( 2 , ) for which 2 € 8 , then the function E is a resolution of the identity for T and the operational calculus is given by the formula ( vi ) ...
Page 891
Nelson Dunford, Jacob T. Schwartz. scalar function f with respect to the operator valued set function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of the integral will be ...
Nelson Dunford, Jacob T. Schwartz. scalar function f with respect to the operator valued set function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of the integral will be ...
Page 951
... f is 2 - measurable , then the function f ( x - y ) is a λx - measurable function . ( b ) For f , ge L1 ( R ) the function f ( x − y ) g ( y ) is integrable in y for almost all x and the convolution f✩g of f and g , which is defined by ...
... f is 2 - measurable , then the function f ( x - y ) is a λx - measurable function . ( b ) For f , ge L1 ( R ) the function f ( x − y ) g ( y ) is integrable in y for almost all x and the convolution f✩g of f and g , which is defined by ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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A₁ 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(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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero