On the differentiability and the representation of one-parameter semi-group of linear operators

Kôsaku YOSIDA

doi:10.2969/jmsj/00110015

Kôsaku YOSIDA

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ジャーナルフリー

1948 年 1 巻 1 号 p. 15-21

DOI https://doi.org/10.2969/jmsj/00110015

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Published: 1948 Received: 1947/10/25 Available on J-STAGE: 2006/08/29 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/08/29 訂正理由: - 訂正箇所: 著者情報訂正内容: Wrong : Kosaku YOSIDA¹⁾
Right : Kôsaku YOSIDA¹⁾

訂正日: 2006/08/29 訂正理由: - 訂正箇所: 所属機関情報訂正内容: 訂正前 :
1) Mathematical institute, Nagoya University

訂正後 :
1) Mathematical Institute, Nagoya University

訂正日: 2006/08/29 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) Cf. I. Gelfand: C. R. URSS, 25 (1939), 713-718 and N. Dunford-I. E. Segal: Bullet. Amer. Math. Soc, 52 (1946), 911-914.
2) According to N. Dunford and I. E. Sepal's paper referred to in 1), E. Hille (Proc, Nat. Acad. Sci., 28 (1942), 175-178, 421-424 obtained the representation U_ix=lim<n→∞> exp (tn(U_1/n-I))x. Hille's parer is not available to the author.
3) The question of the characterization of A together with differentiability of U_i is proposed to the author by Dr. K. Ito in connection with his theory of stochastic differential equations. See his forthcoming paper in Jap. J. of Math.
4) Proc. Nat. Acad. Sei., 16 (1930), 172-175. See also J. von Neumann: Ann. of Math., 33 (1932), 567-573. Proofs of Stone's theorem are given by many authors: F. Riesz, B. von Sz. Nagy and II. Nakano.
5) S. Bochner: Fund. Math., 20 (1933), 262-276. G. Birkhoff: Trans. Amer. Math Soc. 38 (1935), 357-378. I. Gelfand: Commun. Inst. Sci, Math. et Mech. Univ. Kharkoff, 13 (1936), 35-40. B. J. Pettis: Trans. Amer. Meth. Soc., 44 (1938), 277-304.
6) In the case of Stone's theorem, we may replace (1•3) by the separability of { U_ix;-∞<t<∞} and the weak measurability of U_i. This may be carried out by virtue of N. Dunford's theorem in Ann. of Math., 39 (1938), 567-573. This fact is, however, already proved by J. von Neumann in another way. See his paper referred to in 4).

Right : 1) Cf. I. Gelfand: C. R. URSS, 25 (1939), 713-718 and N. Dunford-I. E. Segal: Bullet. Amer. Math. Soc, 52 (1946), 911-914.
2) According to N. Dunford and I. E. Segal's paper referred to in 1), E. Hille (Proc, Nat. Acad. Sci., 28 (1942), 175-178, 421-424 obtained the representation U_tx=lim_n→∞ exp (tn(U_1/n-I))x. Hille's parer is not available to the author.
3) The question of the characterization of A together with differentiability of U_t is proposed to the author by Dr. K. Itô in connection with his theory of stochastic differential equations. See his forthcoming paper in Jap. J. of Math.
4) Proc. Nat. Acad. Sci., 16 (1930), 172-175. See also J. von Neumann: Ann. of Math., 33 (1932), 567-573. Proofs of Stone's theorem are given by many authors: F. Riesz, B. von Sz. Nagy and H. Nakano.
5) S. Bochner: Fund. Math., 20 (1933), 262-276. G. Birkhoff: Trans. Amer. Math Soc. 38 (1935), 357-378. I. Gelfand: Commun. Inst. Sci, Math. et Mech. Univ. Kharkoff, 13 (1936), 35-40. B. J. Pettis: Trans. Amer. Meth. Soc., 44 (1938), 277-304.
6) In the case of Stone's theorem, we may replace (1·3) by the separability of { U_ix;-∞<t<∞} and the weak measurability of U_i. This may be carried out by virtue of N. Dunford's theorem in Ann. of Math., 39 (1938), 567-573. This fact is, however, already proved by J. von Neumann in another way. See his paper referred to in 4).

訂正日: 2006/08/29 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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