Linear Operators: Spectral theory |
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Page 891
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 restricted to that case. Let X be a field of subsets
of a set ...
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 restricted to that case. Let X be a field of subsets
of a set ...
Page 1178
Then 3% maps scalar-valued functions into functions with values in l,. It is plain
from Plancherel's theorem that % is a bounded mapping of the space Le of scalar
-valued functions into the space La(le) of square-integrable vector-valued ...
Then 3% maps scalar-valued functions into functions with values in l,. It is plain
from Plancherel's theorem that % is a bounded mapping of the space Le of scalar
-valued functions into the space La(le) of square-integrable vector-valued ...
Page 1179
By Plancherel's theorem, 4° is a bounded mapping of L2(l,) into the space of
scalar-valued functions le. Thus, by Corollary 19 and Corollary 17, 2 is a
bounded mapping of L,(la) into L,. It is clear from (63) and (61) that %.4 maps G
into the ...
By Plancherel's theorem, 4° is a bounded mapping of L2(l,) into the space of
scalar-valued functions le. Thus, by Corollary 19 and Corollary 17, 2 is a
bounded mapping of L,(la) into L,. It is clear from (63) and (61) that %.4 maps G
into the ...
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Contents
SPECTRAL THEORY | 858 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero