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

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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).