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By Gurarii V. P.

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1 any norm in R"). sup I+(s)l -r 0, we shall let x, denote the element of X defined by s,(s) = x ( t + s ) for - r < s < 0. s, is called the past history of x at t . Let L : X - + R" be a continuous (and so bounded) linear operator mapping X into R". By Riesz representation theorem (see Appendix IV), there exists an n x n matrix n(O), the elements of which are of bounded variation such that ~ ( 4 =) 4Ex [dn(e)] 4(0), (Stieltjes integral). 1) L(x,).

Solve the initial value problems (i) [Wt,t)/dtl + t2du(t, t)/dt where lim,,,u(t, (ii) = 0, t ) = xo(t), uniformly in du(t, t)/dt = Wt, t)/d<, 0 t 2 0, < t < 1, < for sufficiently smooth ~ ~ ( 5 ) ; t 2 0, t 2 0, where Iimt+, u ( ( , t ) = x0(t), uniformly in 5 for sufficiently smooth xo(t). 3. 3. 33 The Hille- Yosida-Phillips Theorem continuous semigroup { T(r)},t >, 0. Recall that the resolvent set p ( A ) of A consists of all complex numbers A for which (AZ-A)-' exists as a bounded operator with domain X .

Claim 7: D ( A ) c D ( B ) and A x = Bx for x the identity = E D(A). 6) (s)B, x ds. 6). To this end observe that for x E D ( A ) IIS,(s)B,x -WAX11 < IIS,(s)II IIB,x - Axll + IICSAW - T(s)lAxll < Mexp(o,s) IIB,x- Axll + 2Mexp(ol s) llAxll + 0 as A + co, uniformly in S on every closed interval [0, t ] . We conclude that T(t)x-x = Hence Bx sb T(s)Axds, = lim 1-0, = = x E D(A). [T(t)x-x]/t lim t - I l T ( s ) A x ds 1-10, AX, and Claim 7 is proved. Claim 8: D ( B ) c D ( A ) . By Claim 6, we have p ( A ) n p ( B ) # I , E p ( A ) n p ( B ) .

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Completeness of systems of shift functions in weighted spaces by Gurarii V. P.


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