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 ( ...
... 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 ( ...
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 1075
... function f in L1 ( —∞ , ∞∞ ) L2 ( − ∞ , + ∞ ) such that the limit in Exercise 12 - fails to exist for x = € 0 . 15 Show that there exists a function f in L1 ( -∞ , ∞ ) for which the family of functions f ( x ) = 1 2π [ ** F ( t ) e ...
... function f in L1 ( —∞ , ∞∞ ) L2 ( − ∞ , + ∞ ) such that the limit in Exercise 12 - fails to exist for x = € 0 . 15 Show that there exists a function f in L1 ( -∞ , ∞ ) for which the family of functions f ( x ) = 1 2π [ ** F ( t ) e ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero