Singular integral operators on analytic or harmonic function spaces and prediction theory

1.设a，b，c，d为复数。设A，B是Hilbert空间H上的有界线性算子，x表示两个子空间的夹角，A的值域和B的值域，y表示两个算子A和B的夹角，即cos y=sup{|&lt；Af，Bf&gt；|；‖Af‖=‖Bf‖=1，f∈H}.然后我们用x或y估计H上的算子范数‖(AA+BB)(CA+dB)^&lt；-1&gt；‖.如果我们用y，则我们可以估计无界算子A和B的范数(cf.5.关于数值范围和半径的讲习班(希腊纳夫普利奥)单位圆上的Helson-Szego权W出现在预测论和函数论中。然后是Log W∈BMO(有界平均振荡函数空间)。有许多实函数υ，如esssup|υ|&lt；π/2和esssup|log W-Hυ|&lt；∞，其中H表示希尔伯特变换。对于给定的W，所有υ的集合C_W是凸集。我们利用单位圆盘上有界解析函数空间的闭单位球来参数化C_W。2000年国际数学分析及其应用会议(台湾高雄)。设P(≠0，I)是满足P^2=P的有界线性算子，Q=I-P，a，b是复数。我们估计了X上的算子范数‖Ap+Bq‖。作为应用，我们用二维情形的结果证明了当X是Hilbert空间时的Feldman-Krupnik-Markus公式。设W&gt；0是单位圆上的可积函数。设P表示从加权勒贝格空间L^2(W)到加权Hardy空间H^2(W)的解析(Riesz)投影，Q=I-P。对于单位圆上的两个有界可测函数a，b，我们研究了L^2(W)上的奇异积分算子AP+Bq。我们建立了算子范数‖Ap+Bq‖的三种范数公式(参见Takahiko Nakazi和Takanori Yamamoto，加权L^2空间上某些奇异积分算子的范数，预印本1-28)。

1. Let a, b, c, d be complex numbers. Let A,B be bounded linear operators on a Hilbert space H.Let x denote the angle between two subspaces, the range of A and the range of B.Let y denote the angle between two operators A and B.That is, cos y = sup{|< Af, Bf >| ; ‖Af‖=‖Bf‖=1, f∈H}. Then we estimate the operator norm ‖(aA + bB)(cA+dB)^<-1>‖ on H using x or y. If we use y, then we can esitimate the norm for unbounded operators A and B (cf. 5th Workshop on Numerical Ranges and Radii (Nafplio, Greece)).2. Helson-Szego weight W on the unit circle appears in the prediction theory and the function theory. Then log W ∈ BMO (the space of functions of bounded mean oscillation). There are many real function υ such that esssup|υ|< π/2 and esssup|log W - Hυ|<∞, where H denote the Hilbert transform. For a given W, the set C_W of all υ is a convex set. We parametrize C_W using the closed unit ball of the space of bounded analytic functions on the unit disc (cf. International Conference on Mathematical Analysis and its Applications, 2000 (Kaohsiung, Taiwan)).3. Let P(≠0, I) be a bounded linear operator satisfying P^2=P, and let Q=I-P.Let a, b be complex numbers. We estimate the operator norm ‖aP + bQ‖ on X.As an application, we use the result for 2-dimensional case to prove the formula of Feldman-Krupnik-Markus when X is a Hilbert space.4. Let W > 0 be an integrable function on the unit circle. Let P denote the analytic (Riesz) projection from the weighted Lebesgue space L^2 (W) to the weighted Hardy space H^2 (W), and let Q=I-P.For two bounded measurable functions a, b on the unit circle, we studied the singular integral operators aP+bQ on L^2 (W). We established three kinds of norm formulas for the operator norm ‖aP+bQ‖ (cf.Takahiko Nakazi and Takanori Yamamoto, Norms of some singular integral operators on weighted L^2 spaces, preprint 1-28).