Linear Operators: Spectral theory |
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Page 890
for which 2 , ed , then the function E is a resolution of the identity for T and the operational calculus is given by ... integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied .
for which 2 , ed , then the function E is a resolution of the identity for T and the operational calculus is given by ... integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied .
Page 951
( a ) If f is 2 - measurable , then the function f ( x − y ) is a 2x2 - measurable function . ( b ) For f , geLy ( 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 ...
( a ) If f is 2 - measurable , then the function f ( x − y ) is a 2x2 - measurable function . ( b ) For f , geLy ( 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 ...
Page 1680
F ( p ) = S , 7 ( x ) f ( x ) dx , pe C ( 1 ) . It is often suggestive to write F ( ) in this form even in the general case ; that is , to introduce for each F in D ( I ) an ideal " function ” | which is defined by formula ( * ) .
F ( p ) = S , 7 ( x ) f ( x ) dx , pe C ( 1 ) . It is often suggestive to write F ( ) in this form even in the general case ; that is , to introduce for each F in D ( I ) an ideal " function ” | which is defined by formula ( * ) .
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
Copyright | |
44 other sections not shown
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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero