Gap between the induced norm of a matrix and largest Eigenvalue? Are there any known results on how big the gap between the absolute value of the largest Eigen  value of matrix and the induced norm can be?

More formally, let the induced norm of A is given by $\|A\| = max_{\|x\| = 1}\|Ax\|$ and let $\lambda_{max}$ denote the largest Eigenvalue. I am interested in the quantity $\|A\| - |\lambda_{max}|$. Since the norm bounds the absolute value of the Eigenvalues, the quantity $\|A\| - |\lambda_{max}|$ is always positive. I also know that for positive-definite matrices, the quantity $\|A\| - |\lambda_{max}|$ is zero. But are there any results known for a generic matrix $A$?

p.s: I am working on a problem where I am trying to compute a bound on the norm of a matrix. I have bounds on the Eigenvalues of my matrix via Gresghorin's circle theorem and I am trying to see whether I can use that in some way to obtain a bound on the matrix norm.... 

Edit: To clarify, A is a square matrix over the field of reals and I am using the standard 2-norm on $R^n$
Presumably you are talking about the induced 2-norm, i.e. the largest singular value. When the spectral radius of $A$ is fixed, the 2-norm can be unbounded, as illustrated by the example $A=\pmatrix{1&amp;n\\ 0&amp;1}$. We have $\rho(A)=1$ but $\|A\|_2$ (and in turn the gap $\|A\|_2-\rho(A)$) approaches infinity when $n\to\infty$.
Your two notions of norm are equal. If the maximum absolute value eigenvalue of $A$ is $\lambda$ and you take any vector $w$ of length $1$, then you will have $||Aw|| \leq |\lambda| ||w|| = |\lambda|$, and you will have equality when you take $w$ to be an eigenvector of $A$ with eigenvalue $\lambda$.