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!