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!



Operational quantities characterizing strictly singular operators and strictly cosingular operators
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 $\delta sin'<sq'$.


