# Operational quantities characterizing strictly singular operators and strictly cosingular operators

An operational quantity is a procedure which determines, for every pair $X,Y$ of infinite-dimensional Banach spaces, a map from $\mathcal{L}(X,Y)$ into the non-negative numbers.

Given two operational quantities $a$ and $b$, we will write $a\leq b$ if for any infinite-dimensional Banach spaces $X,Y$ and $T\in \mathcal{L}(X,Y)$, we have $a(T)\leq b(T)$. We will say that $a$ and $b$ are equivalent if $\alpha a\leq b\leq \beta a$ for some $\alpha, \beta>0$.

M. Schechter (Quantities related to strictly singular operators, Indiana Univ. Math. J. 1972) introduced two quantities measuring strictly singular operators as follows:

For an operator $T\in\mathcal{L}(X,Y)$, we set

$$\Delta(T)=\sup_{M}\inf_{N\subseteq M}\|T|_{N}\|,$$ where $M,N$ are infinite-dimensional closed subspaces of $X$.

$$\tau(T)=\sup_{M}\inf_{x\in S_{M}}\|Tx\|,$$ where $M$ is an infinite-dimensional closed subspace of $X$.

It is easy to see that $\tau\leq \Delta$. But it seems that he did not show that these two quantities are not equivalent.

Question 1. Are quantities $\tau$ and $\Delta$ equivalent?

J. Zemanek (Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour, Studia Math, 1984) introduced two quantites measuring strictly cosingular operators as follows:

For an operator $T:X\rightarrow Y$, the surjection modulus of $T$ is defined as

$$q(T)=\sup\{\epsilon\geq 0: \epsilon B_{Y}\subseteq TB_{X}\}.$$

we set

$$\nu(T)=\sup\{q(Q_{V}T): codim V=\infty\},$$ where $Q_{V}:Y\rightarrow Y/V$ is the quotient map.

$$\nabla(T)=\sup_{W}\inf_{W\subseteq V}\|Q_{V}T\|,$$ where $W,V$ are infinite-codimensional closed subspaces of $Y$.

It is easy to see that $\nu\leq \nabla$.

Question 2: Are quantites $\nu$ and $\nabla$ equivalent?

Thank you!

• You may want to check out the paper "Operational quantities characterizing semi-Fredholm operators" by Gonzalez and Martinon. They discuss some related operational quantities and use distortable Banach spaces to give examples of their nonequivalence. These examples may be adapted to the quantities you are looking at here. Nov 25, 2016 at 18:40
• Yes, the non equivalence of these quantities for operators from $\ell_2$ follows from the Odell-Schlumprecht solution of the distorted norm problem. For general Banach spaces I think you can derive it from Tsirelson's construction of his space and Milman's work in the 1960s on distortion. Nov 25, 2016 at 21:06
• This is all very interesting, by the way. I have never encountered operational quantities before. They seem to be closely related to operator ideals, and I wonder if they could be used to solve some of the open problems of the form "Does $\mathcal{B}(X)$ admit infinitely many closed ideals?" say for $X=L_1$, $\ell_\infty$, or $\ell_p\oplus c_0$. Nov 25, 2016 at 21:19
• I guess I am curious to know the following: If $a$ is an operational quantity, denote $ker(a)=$ the class of all operators $T$ for which $a(T)=0$. There are several known choices of $a$ for which $ker(a)$ is the class of strictly singular operators. However, has anyone ever studied choices of $a$ for which $ker(a)$ is a norm-closed operator ideal which is not very well-known---i.e., not the compact/weakly compact/Rosenthal/strictly singular/inessential/etc.---? In particular it would be interesting to use operational quantities to construct an entirely novel norm-closed operator ideal. Nov 25, 2016 at 21:36
• @BenWallis I am recently doing some research on operational quantites related to strictly singular operators and strictly cosingular operators. If you are interested in this topic, we can work together. Nov 26, 2016 at 13:49

The quantities $\Delta$ and $\tau$ are better known in the literature as $sin$ and $sj$, respectively. In the paper "Note on Operational Quantities and Mil'man Isometry Spectrum" by Gonzalez/Martinon (Rev. Acad. Canar. Cienc. 3, 1991, pp103–111), it is shown that $sin$ and $sj$ fail to be equivalent---in particular, there is no $\delta>0$ such that $\delta sin<sj$.
Now, the quantities $\nu$ and $\nabla$ are better known as $sq'$ and $sin'$, respectively. In "Operational quantities characterizing the semi-Fredholm operators" (Studia Math. 114, 1995, pp13–27), also by Gonzalez/Martinon, it is shown that these fail to be equivalent---in particular, there is no $\delta>0$ such that $\delta sin'<sq'$.
• Thanks, Ben. I do not know whether the counterexamples to the non-equivalence of $sin$ and $sj$ can be used for quantities $\Delta$ and $\tau$. Nov 26, 2016 at 13:57
• In "Operational quantities the semi-Fredholm operators"(Studia Math. 114, 1995, 13-27), I can not find any result showing that the quantities $\nu$ and $\nabla$ are non-equivalent. Dec 10, 2016 at 8:51