$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$ Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
  $$ \|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}\,, $$
where $\|T\|_p$ are the induced operator norms of $T$.

  
*
  
*Does this inequality have a name?
  
*What does the ratio $\sqrt{\|T\|_1\|T\|_\infty}\,\big/\,\|T\|_2$ tell us about $T$?
  Has it ever been studied? Is there any standard notation or name for it?
  

 A: Sorry, my answer below is only partial, but I thought that it may still be somewhat interesting.

As far as I know, this inequality does not have a distinguished name. It is ultimately a consequence of the duality between $\|\cdot\|_\infty$ and $\|\cdot\|_1$, and the submultiplicativity of $\|\cdot\|_1$.
This is certainly known to you, but I wanted to remark that $$\|T\|_2^2 = \|T^*T\|_2 = \rho(T^*T) \le \|T^*T\|_1 \le \|T^*\|_1 \|T\|_1 = \|T\|_\infty \|T \|_1.$$ Thus, this inequality is nothing but an expression of: 


*

*The C*-identity of the spectral norm

*Dominance of the spectral radius $\rho(\cdot)$ by the $\|\cdot\|_1$-norm.

*Submultiplicativity of the operator $\|\cdot\|_1$-norm.


As to the second inequality in the OP, we might need some further restrictions to make it more interesting. For instance, if $T$ is symmetric, then it reduces to $\|T\|_2 \le \|T\|_1$, whereby the ratio becomes $\|T\|_1/\|T\|_2$. This ratio has been studied for vectors (diagonal $T$) in the world of "sparse coding / compressed sensing, etc." --- see e.g., the paper here, and references therein. 
A: In this paper, Ron DeVore refers to this inequality as simply "interpolation of operators." He uses it to prove that a particular matrix $\Phi$ satisfies the restricted isometry property, i.e., that every sufficiently small principal submatrix $A$ of $\Phi^*\Phi$ has all of its eigenvalues between $1/2$ and $3/2$, or equivalently, $\|A-I\|_2\leq1/2$. Of course, DeVore appeals to interpolation of operators since $\|A-I\|_1$ and $\|A-I\|_\infty$ are easier to access.
I don't know if the ratio you mentioned has been studied before, but it certainly measures how tight the bound is. It's worth mentioning that in the case where the diagonal of $T$ is all zeros, interpolation of operators follows from the Gershgorin circle theorem, and these are actually equivalent if $T$ is also self-adjoint. In this case, this work might be related to your ratio.
