Exponential-cosine operator-valued functional equation in the strong operator topology |
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Authors: | Harshinder Singh H L Vasudeva |
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Institution: | (1) Department of Statistics and Mathematics, Panjab University, Chandigarh, India |
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Abstract: | LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR
+ = 0, ). A one parameter family of operators {S(t);t R
+},S:R
+ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S
2(s))S(t –s), for alls, t R
+,s t. Let
,fD(A), and
,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A
2) implies
0
t
(t –u(S(u)fduD(B) and B
0
t
(t –u)S(u)fdu =S(t)f –f +tAf – 2A
0
t
S(u)fdu + 2A
2
0
t
(t –u)S(u)fdu.D(B) is seen to be dense inD(A
2). Some regularity properties ofS(t) have also been obtained. |
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Keywords: | Primary 39B70 47D05 |
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