sesquilinearform only small on diagonal

Question

We are given a sesquilinear form $a \colon V \times V \to \mathbb{C}$ on a complex Hilbert space $V$. We know that $a$ is bounded and $a(u,u)$ is small for all $u \in V$. Can we make any assertions on the size of $a(u,v)$, that is to say, can we (under further assumptions?) make assertions about the size of a sesquilinear form if it it small on the diagonal in $V\times V$?

Answer 1

By assumption $|a(u,u)|\leq c\lVert u\rVert^2$ for all $u$, where $c>0$ is a constant.

Considering the polarization formula, we have

$$|a(u,v)|\leq\frac14\Big(c\lVert u+v\rVert^2+c\lVert u-v\rVert^2+c\lVert u+iv\rVert^2+c\lVert u-iv\rVert^2\Big)$$ $$=\frac14\Big(c\big(\lVert u\rVert^2+\langle u,v\rangle+\langle v,u\rangle+\lVert v\rVert^2\big)+c\big(\lVert u\rVert^2-\langle u,v\rangle-\langle v,u\rangle+\lVert v\rVert^2\big)\qquad \\ \qquad+c\big(\lVert u\rVert^2+\langle u,iv\rangle+\langle iv,u\rangle+\lVert v\rVert^2\big)+c\big(\lVert u\rVert^2-\langle u,iv\rangle-\langle iv,u\rangle+\lVert v\rVert^2\big)\Big)$$ $$=\frac14\Big(4c\lVert u\rVert^2+4c\lVert v\rVert^2\Big)$$ $$=c\Big(\lVert u\rVert^2+\lVert v\rVert^2\Big).$$

Now, given any $u,v\neq0$, let $u'=\sqrt\frac{\lVert v\rVert}{\lVert u\rVert}u$ and $v'=\sqrt\frac{\lVert u\rVert}{\lVert v\rVert}v$, so that $\lVert u'\rVert=\lVert v'\rVert=\sqrt{\lVert u\rVert\lVert v\rVert}$, and

$$|a(u,v)|=|a(u',v')|\leq c\Big(\lVert u'\rVert^2+\lVert v'\rVert^2\Big)=2c\lVert u\rVert\lVert v\rVert.$$