A question in Banach space

Question

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively.

Where could I find a reference for fact that $$\gamma_2^2(A)\leq\| A\|_{\ell_1^n\rightarrow\ell_{\infty}^m}rank(A)?$$

Should this be $$\gamma_2^2(A)\leq\| A\|_{\ell_1^n\rightarrow\ell_{\infty}^m}^2rank(A)?$$

Above is lemma 4.2 in http://www2.mta.ac.il/~adish/Pubs/Papers/complexity_matrices.pdf

What is terminology for $\gamma_2(A)$ in literature?

posted https://math.stackexchange.com/questions/1196950/a-question-in-banach-space

Answer 1

Given a linear operator $A$ from a Banach space $E$ to a Banach space $F$, the quantity $\gamma_2(A)$ is the infimum (actually, it is a minimum) of $\|X \| \cdot \|Y\|$, where the minimum is over all operators $Y: E\to H$, $X: H \to F$ s.t. $XY=A$ with $H$ a Hilbert space. This agrees with your definition when $E= \ell_1^n$ and $F= \ell_\infty^m$. The quantity $\gamma_2(\cdot)$ is an important operator ideal norm; it is discussed in many books on Banach space theory. The most elementary of those is the book of Albiac and Kalton (section 7.3), but they do not use the notation $\gamma_2(\cdot)$. Or look at Diestel, Jarchow, Tonge or Tomczak-Jaegermann. The second form for your inequality is the correct one (after all, it must be homogeneous) and is immediate from the ideal property of $\gamma_2(\cdot)$ and John's theorem that the isomorphism constant of an $n$-dimensional Banach space to $\ell_2^n$ is at most $\sqrt{n}$.