TAILIEUCHUNG - Báo cáo toán học: "Curvature inequalities and extremal properties of bundle shifts "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Cong bất bình đẳng và tài sản cực trị của ca bó. | Copyright by INCREST 1984 J. OPERATOR THEORY 11 1984 305 -317 CURVATURE INEQUALITIES AND EXTREMAL PROPERTIES OF BUNDLE SHIFTS GADADHAR MISRA INTRODUCTION We study a class of Hilbert space operators Bx i2 possessing an open set of eigenvalues Í2. The eigenspaces of an operator T in Bỵịữ form a complex holomorphic hermitian line bundle over Í2 defined as ET co x - co co x e ĨĨ X and Tx cox . One reason for studying the class B íĩ is the following theorem due to Cowen and Douglas 8 . Theorem. Two operators T andT in Bffi are unitarily equivalent if and only if. r co c ỹ co where 7 co dco A dco is the curvature of the bundle ET . We estimate the curvature of an operator T in Bffi using the Szego kernel for Í2 whenever C1Í2 is a spectral set for T. These estimates are then shown to be best possible. The class Bt Q was first introduced in Cowen and Douglas 8 as follows B Q T 6 ựd i Q G a T ii ran T co 2 co e Í iii V ker T co 2 wen iv dimker T co 1 . It is important for US to relate the curvature of the bundle ET to the operator T. The space ker T co 2 is invariant for T co and T co I ker T co 2 is a nonzero nilpotent operator of order two. Relative to an appropriate basis we have the matrix representation Na TI ker T - co 2 r _0 co Now T co bears a simple relationship to hT co namely xr co 7r co _2. Our problem is to find reasonable estimates for operators in Bffi . If clD is a spectral set for T then it is also a spectral set for Na. We show that C1Í2 is a 306 G. MISRA. spectral set for Na if and only if ỉr ft ế Kfí ừ ft -1 where ft is the Szego kernel for the region ÍÌ. In turn we obtain the desired curvature inequality. At this point it is natural to ask if C1Í2 has to be a spectral set for T whenever T satisfies the curvature inequality. We are not able to resolve this question. However see note at the end. To show that the estimate T co Kq co cũ 2 is best possible we must establish the existence of an operator T in 5j i2 satisfying jfr ft ft 2. First we attempt to

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