Linear Operators, Part 2 |
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Page 1338
( ii ) we have Misc Üem ) = Erslem ) m = 1 mel for each sequence of disjoint Borel
sets with bounded union . 7 LEMMA . Let { uj } be a positive matrix measure
whose elements Mis are continuous with respect to a positive o - finite measure u
.
( ii ) we have Misc Üem ) = Erslem ) m = 1 mel for each sequence of disjoint Borel
sets with bounded union . 7 LEMMA . Let { uj } be a positive matrix measure
whose elements Mis are continuous with respect to a positive o - finite measure u
.
Page 1340
Let û be another o - finite positive regular measure with respect to which the set
functions Mis are continuous . Let { ij } be the corresponding matrix of densities ,
and let ( nu } be the matrix of densities of the Mi ; with respect to the measure ( u +
...
Let û be another o - finite positive regular measure with respect to which the set
functions Mis are continuous . Let { ij } be the corresponding matrix of densities ,
and let ( nu } be the matrix of densities of the Mi ; with respect to the measure ( u +
...
Page 1738
... e CO ( E ? ) , the function y defined by 4 . ( x ) = 56 ( X1 ) q ( X2 , . . . , ) belongs
to C ( V ) , and Sq ( x1 ) = æ ? - for all sufficiently small dz . Plainly yoV is in CP ( V
+ ) . It follows on integrating by parts p times with respect to ty that ( * ) Sv h ( x ) ...
... e CO ( E ? ) , the function y defined by 4 . ( x ) = 56 ( X1 ) q ( X2 , . . . , ) belongs
to C ( V ) , and Sq ( x1 ) = æ ? - for all sufficiently small dz . Plainly yoV is in CP ( V
+ ) . It follows on integrating by parts p times with respect to ty that ( * ) Sv h ( x ) ...
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Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero