REMARKS ON A HILBERT SPACE OF ANALYTIC FUNCTIONS.
REMARKS ON A HILBERT SPACE OF ANALYTIC FUNCTIONS.
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关于解析函数希尔伯特空间的评论。
DOI：
10.1073/pnas.48.2.199
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发表时间：
1962
影响因子：
11.1
通讯作者：
V. Bargmann
中科院分区：
文献类型：
--
作者：
V. Bargmann
遍历定理*由亚历山德拉·伊奥内斯库·图尔恰和卡修斯·伊奥内斯库·图尔恰宾夕法尼亚大学，1961年12月21日由埃纳·希尔发表。设（Z.）是一个完备的全有限测度空间，E是一个巴拿赫空间。对于每一个1 < p < 0用4C表示。所有（Bochner）可测映射f(Z)到E的向量空间，其中Z (Z)|P是4可积的；这里给JE赋以HfK||的半范数p = (Jzilf(z)jjPdAt(z))l/ p，用LE表示相关的分离（Banach）空间，用f -f表示4e到LE的正则映射。设SE为所有有界且属于£E的函数的向量空间；如果| = ess SupzIZ-Zf(z) j。用SE表示相关联的分隔（赋范）空间，用f表示SE到SE的正则映射。设D为SE到SE的所有线性映射T‘的集合，满足’ |TI1, < I and t11 '11。< 1。然后||T||p < 1对于所有1 < p < 0；因此，T可以通过连续性扩展到L'（我们用同一个字母表示扩展）。对于T E D和f ' o = U1 Up<=，我们用Tf a（确定的）表示Tf类的代表。Let To, T1，…，柯度；考虑条件：(1)To = I；(2) TjTj = TjTj，对于i，ji{0。1 . .。1 k;(3) TjTj+1 = Tj+1，对于j E{0,1，…， k 1}。我们定义To = I为allj E {O， 1，…k。对于每个函数E V和a >，设Gf(a) = {z |||f(z)H> a}。定理1。让To， T1，…， Tk E D为k + 1个满足条件(1)、(2)、(3)的算子。对于每个ea和每个b>，定义G*(a) = Iz Supr oil 0,1。k), N II(Tq + Tj±…+Tj)f(z)/(n + 1)|I> a}。然后，对于每个集合F E g验证关系Gf(a) c F c Gf(a)我们有aM(F) <。Fpjf(z)jld1A(z) < c2设T E d，对于每一个E u和每一个a b b 0 0，定义E；(a) = {zJ Sup8ENJI(TO + T1 +…+ T')f(z)/(n + 1) || > al，则a/I(Ef(a)) <。Ef (a) If(Z) IdA(Z) <推论1由定理1导出，取k = 1, To = I, T = T, F = E；(一)。
ERGODIC THEOREMS* BY ALEXANDRA IONESCU TULCEA AND CASSIUS IONESCU TULCEA UNIVERSITY OF PENNSYLVANIA Communicated by Einar Hille, December 21, 1961 1. Let (Z. £, , be a complete totally a-finite measure space and E a Banach space. For each 1 < p < o denote by 4C. the vector space of all (Bochner) measurable mappings f of Z into E for which z oflf(z)|P is 4-integrable; here JE is endowed with the semi-norm f HfK||p = (Jzilf(z)jjPdAt(z))l/P Denote by LE the associated separated (Banach) space and by f -f the canonical mapping of 4e onto LE. Let SE be the vector space of all functions which are bounded and belong to £E; here SE is endowed with the semi-norm f ||If| = ess SupzIZ-Zf(z) j. Denote by SE the associated separated (normed) space and by f f the canonical mapping of SE onto SE. Let D be the set of all linear mappings T' of SE into SE such that' |TI1, < I and T1l'11. < 1. Then ||T||p < 1 for all 1 < p < o; hence, T can be extended by continuity to L' (we denote the extension by the same letter). For T E D and f ' o = U1 Up<=, 2P we denote by Tf a (determined) representative of the class Tf. 2. Let To, T1, . .. , TkE DU; consider the conditions: (1) To = I; (2) TjTj = TjTj for i,ji {O. 1,.. ., 1k; (3) TjTj+1 = Tj+1 for j E {O, 1, . . . , k 1}. We define To = I for allj E {O, 1, . . .,k, . For each functionf E V and each a > O let Gf(a) = {z |||f(z)H> a}. THEOREM 1. Let To, T1,.. ., Tk E D be k + 1 operators satisfying the conditions (1), (2), (3). For eachf E a and each a > 0, define G*(a) = Iz Supr oiI0,1. k), N II(Tq + Tj ± ... +Tj)f(z)/(n + 1)|I> a}. Then, for each set F E g verifying (except for sets of measure zero) the relations Gf(a) c F c Gf(a) we have aM(F) <. Fpjf(z)jld1A(z) < c2 COROLLARY 1. Let T E D. For eachf E VU and each a > 0, define E;(a) = {zJ Sup8ENJI(TO + T1 + ... + T')f(z)/(n + 1) || > al. Then, a/I(Ef(a)) < . Ef (a) If(Z) IdA(Z) < CCorollary 1 follows from Theorem 1 if we take k = 1, To = I, T. = T and F = E; (a).