Triangular truncation and normal limits of nilpotent operators

Don Hadwin
1995 Proceedings of the American Mathematical Society  
We show that, as n -» oo , the product of the norm of the triangular truncation map on the n x n complex matrices with the distance from the normone hermitian «x« matrices to the nilpotents converges to 1/2. We also include an elementary proof of D. Herrero's characterization of the normal operators that are norm limits of nilpotents. Suppose zz is a positive integer and let Jin , ETn , jVn denote, respectively, the sets of all zz x zz complex matrices, strictly upper triangular zz x zz
more » ... , and nilpotent zz x zz matrices. There is a natural mapping x" : Jin -*■ &ñ > namely, x"(T) replaces the entries on or below the main diagonal of T with zeroes. The map x" is called triangular truncation on Jfn . On an infinite-dimensional space, the triangular truncation mapping does not always yield the matrix of a bounded operator. This is related to the fact that the range of the mapping that sends a bounded harmonic function on the unit disk to its analytic part is not included in H°° . For example, if f(z) = log(l -z), then u = 2ilm(f) is bounded in modulus by n, but the analytic part of u, namely /, is not bounded. In terms of Toeplitz operators, Tu is an operator with norm n , but the upper triangular truncation of Tu is the formal matrix for Tf, which is not a bounded operator. The matrix for Tu is the matrix whose (i, z')-entry is 0 and (i, »-entry is l/(j -i) for 1 < i ^ j < oo. For each positive integer zz, let TUy" be the zz x zz upper-left-hand corner of Tu , i.e., the (z, z')-entry of Tu¡" is 0, and the (i, »-entry of Tu" is l/(j-i) for 1 < i # j n|| < n for each zz, and that ||T"(r",")|| -> oo as zz -► oo. Hence ||t"|| -»oc as zz -> oo . Much work has been done in determining ||r"||. S. Kwapien and A. Pelczynski [KP, pp. 45-48] proved in 1970 that ||t"|| = 0(log(zz)), K. Davidson [D, p. 39] proved that ^ < liminf"_00 ||T"||/log(zz), and, in 1993, J. R. Angelos, C. Cowen, and S. K. Narayan [ACN] proved that lim ||T"||/l0g(«) = 1/7T.
doi:10.1090/s0002-9939-1995-1257109-0 fatcat:qmktrzqipvcnhcz5x7tq5chll4