Linear Operators: Spectral theory |
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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 1178
... functions boundedly into the space L , ( 2 ) of vector- valued functions . n Let M be the mapping in L , ( l ) which maps the vector - valued function whose nth component has the Fourier transform ( § ) into the vector - valued function ...
... functions boundedly into the space L , ( 2 ) of vector- valued functions . n Let M be the mapping in L , ( l ) which maps the vector - valued function whose nth component has the Fourier transform ( § ) into the vector - valued function ...
Page 1645
... functions , but can only be a " function " in some generalized sense . Hence , we are led to the attempt to define some sort of " generalized function . " A very complete and interesting development of such a theory of generalized functions ...
... functions , but can only be a " function " in some generalized sense . Hence , we are led to the attempt to define some sort of " generalized function . " A very complete and interesting development of such a theory of generalized functions ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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59 other sections not shown
Common terms and phrases
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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero