Linear Operators: Spectral theory |
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Page 865
An element « in a B - subalgebra of the form Xo = e.Xe , where e , is an idempotent with 0 #lo * e clearly has 0 ( r ) Co ( x ) . The following lemma shows that the opposite inclusion holds in case X , has the same unit as X. 9 LEMMA .
An element « in a B - subalgebra of the form Xo = e.Xe , where e , is an idempotent with 0 #lo * e clearly has 0 ( r ) Co ( x ) . The following lemma shows that the opposite inclusion holds in case X , has the same unit as X. 9 LEMMA .
Page 877
Then an element y in Y has an inverse in X if and only if it has an inverse in y . y Consequently the spectrum of y as an element of y is the same as its spectrum as an element of X. Proof . If y - l exists as an element of Y then ...
Then an element y in Y has an inverse in X if and only if it has an inverse in y . y Consequently the spectrum of y as an element of y is the same as its spectrum as an element of X. Proof . If y - l exists as an element of Y then ...
Page 1339
An element F of Ly ( { Mi ; } ) will be said to be a { M ij } -null function if | F1 = 0. The set of all equivalence classes of elements of Ly ( { uis } ) modulo { uij } -null functions will be denoted by L2 ( { i ; } ) .
An element F of Ly ( { Mi ; } ) will be said to be a { M ij } -null function if | F1 = 0. The set of all equivalence classes of elements of Ly ( { uis } ) modulo { uij } -null functions will be denoted by L2 ( { i ; } ) .
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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