arXiv subjects

Gary Weiss

Publications and source records attributed to Gary Weiss.

21 records · Page 2Linked to original sources

arXiv2007

Soft ideals and arithmetic mean ideals

This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are "soft", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and arithmetic mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal.

math.FA↗

arXiv2007

Second order arithmetic means in operator ideals

Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal ideals for this implication to hold true. We present also sufficient conditions for second order inclusion cancellations. These conditions are formulated in terms of the growth properties of the ratio of regularity sequence associated to the sequence of s-number of a generator of the principal ideal. These results are then extended to general ideals.

math.FA↗

arXiv2007

B(H) lattices, density and arithmetic mean ideals

This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal I is am-stable (I = I_a) if and only if any of its first order arithmetic mean ideals are am-stable if and only if the ideal satisfies the first order equality cancellation property: J_a = I_a implies J = I. We show that this cancellation property can fail even for am-stable countably generated ideals. Similar results hold for arithmetic mean at infinity ideals. Inclusion cancellations do not hold in general even for principal ideals, but for every ideal I there is a largest ideal I^ for which J_a contains I_a implies that J contains I^. When I is principal, I^ too is principal. We show that I=I^ is a strictly stronger property than am-stability. For example, for I the p < 1 power of the principal ideal J generated by diag {1/n}, I^ is the q power of J where 1/p-1/q = 1.

math.FA↗