Download PDF by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.): Analytic Functions Błażejewko 1982: Proceedings of a

By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)

ISBN-10: 3540127127

ISBN-13: 9783540127123

ISBN-10: 3540386971

ISBN-13: 9783540386971

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Additional info for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982

Example text

W)e:UXW. )u,u)Ue:H(D) U=W, for every ue:U. such operator-valued functions is denoted by H(D;B(u:w». )*e:ii(D;B(W:U». H(D;C(U:W». T(')e:H(D;B(U:W» H(D;A(U:U». similar manner. 3. DxD. H(D;Q(U:U}), H(D;C(U:W» operator-valued kernel An sesqui-holomorphic on morphic on i f and only i f DxD i f for any In this case. The class of all The corresponding class is defined analogously. )e:H(DxD;B(U:W» The Basic Theorem Throughout this section and the remaining parts of the paper, D stands for a fixed domain (or a complex manifold) in a:;n.

11) if of all continua in to contain the complement of ~ ball then m y; 1, wn B ! J...! 1 (c) of Vaisala's paper [22J, we have Mpr E M rk k P +E M r m O P m +E M r q 1 P q' where r and r O m =r 1 =¢ q yields M r O =Mp r q1 =0 pm (m, q = 1 ,2, ••• ) • Then, taking into account Lemma 3, it follows that we may suppose, without loss of generality, that D itself is a domain, since otherwise, we can establish (11) for each component of D, separately. The inequality ( 12) Mr P ~ cap (E ,E"D) P 0 can be proved by the same argument as that used by Ziemer [24Jfor the preceding proposition since the additional condition "Co'C, and Co closed containing the complement of a ball" is not involved in the proof, while the use of r X rather simplifies things.

Now, let Z be a o D and let {zm} be a sequence of points of D, converging to z. 5), s(u)(Z) = lim s(u)<> and since u m 0 m-+<><> is an arbitrary unit vector in U, is a Cauchy-sequence for any vector is a Cauchy-sequence for any AEt UEU. and any 0 m we conclude that {(So(zm)u,u)U} In particular, U,WEU. Choosing A=l and A=i, suc- 29 Positive Definiteness and Holomorphy cessively, we conclude that is a Cauchy-sequence for any weakly complete, that weakly in U,WEU. So (Zm) , U to an operator for any S(~)=So(~) ~EDo' S(z)EB(u:u), s(u) (z)EQ, ZED, S(·) ZED.

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Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982 by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)


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