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We show that there are $2^{2^{aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p ot= 2<infty$. This solves a problem in A. Pietschs 1978 book Operator Ideals. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.
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We study the class of functions $f$ on $mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $mathcal{L}_p$ for $0 < p leq 1$. The corresponding problem with $pgeq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})$ obey the estimate $$ |f(A)-f(B)|_{p} leq C_{p}(|f|_{L_{infty}(mathbb{R})}+|f|_{dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})})|A-B|_{p} $$ for all bounded self-adjoint operators $A$ and $B$ with $A-Bin mathcal{L}_p$. In the case $p=1$, our methods recover and provide a new perspective on a result of Peller that $f in dot{B}^1_{infty,1}$ is sufficient for a function to be Lipschitz in $mathcal{L}_1$. We also provide related Holder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on $mathbb{R}$ are not Lipschitz in $mathcal{L}_p$ for any $0 < p < 1$. This gives counterexamples to a 1991 conjecture of Peller that $f in dot{B}^{1/p}_{infty,p}(mathbb{R})$ is sufficient for $f$ to be Lipschitz in $mathcal{L}_p$.
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support? We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author.
A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < infty.
For a general measure space $(Omega,mu)$, it is shown that for every band $M$ in $L_p(mu)$ there exists a decomposition $mu=mu+mu^{primeprime}$ such that $M=L_p(mu)={fin L_p(mu);f=0 mu^{primeprime}text{-a.e.}}$. The theory is illustrated by an example, with an application to absorption semigroups.
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $mathfrak{M}subsetmathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gelfand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator, (ii) a non-closable extension of a symmetric operator and (iii) selfadjoint extensions of Laplacians with a generalized boundary condition.
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