# Lower bounds for norms of commutators

For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound on $||[A,B]||$ in terms of properties of $A$ and $B$. (It is clear that one can not have simple norm estimates since for $A=1$ one has $[A,B]=0$.) One may consider questions like: Is there a subspace $\Omega$ of $B(H)$ such that $||[A,B]||\geq ||A||$ for all $A\in\Omega$ (keeping $B\in B(H)$ fixed)? More generally, I'm also interested in similar questions for the $L^p$-norm $||A||_p=(\operatorname{tr}|A|^p)^{1/p}$. In the literature, one finds a flora of results concerning upper bounds on norms of commutators (however, for the $L^p$-norm, results are sparse). Is anyone familiar with these types of questions, concerning lower bounds? Can someone point out relevant sources?

This doesn't address the full generality of your question but it might at least suggest places to hunt in the literature.

Thinking of $B$ as fixed, you're asking for a lower bound on the norm of $\operatorname{ad}_B(A)$ as $A$ varies in some subspace. If we take $A$ to range over all of $B(H)$ then MathSciNet directs me to

J. G. Stampfli. The norm of a derivation. Pacific J. Math. Volume 33, Number 3 (1970), 737–747

in which it's proved that $\Vert \operatorname{ad}_B \Vert$ is equal to twice the distance of $B$ from the subspace ${\mathbb C} I$.

One can consider similar problems for norms of inner derivations on other C*-algebras (the von Neumann case seems apparently behaves just like $B(H)$, see L. Zsido, The norm of a derivation in a $W^\ast$-algebra, Proc. Amer. Math. Soc. 38 (1973), 147–150). In the unital case, it seems like a good place to find out what's known is

MR2274022 (2008f:46071) R. J. Archbold, D. W. B. Somerset. Measuring noncommutativity in $C^\ast$-algebras. J. Funct. Anal. 242 (2007), no. 1, 247–271.

• Thanks! I will look into these papers and their references. Looks promising! – Joakim Arnlind Mar 6 '14 at 8:04

The following result is classical: let $\mathbb H$ be a Hilbert space, and let $A,B\in \mathcal B(\mathbb H)$, then $[A,B]\not=I.$ In finite dimension, just take the trace, and if the dimension is infinite, compute $[A,B^N]$.

This indicates that the most interesting questions about commutators are dealing with unbounded operators. The paradigmatic examples are the creation ($A_+$) and annihilation operators $(A_-)$ $$A_\pm=D_t\pm i\frac t2,\quad D_t=-i\partial_t.$$ We have $[A_-,A_+]=I,$ which is a way to express the uncertainty principle. $A_+$ is injective with an image of codimension 1 (Fredholm index $-1$) whereas $A_-$ is surjective with a kernel of dimension 1 (Fredholm index $1$). In the Hermite basis, $A_+$ is a forward weighted shift, $A_-$ its adjoint, a backward weighted shift.

An interesting category of unbounded operators is given by pseudodifferential operators: taking $A=a(x,D)$ with order $m_1$, $B=b(x,D)$ with order $m_2$, the principal symbol of the commutator $[A,B]$ is $\frac 1i\{a,b\}$ where $\{a,b\}=\partial_\xi a\cdot \partial_x b- \partial_x a\cdot \partial_\xi b$ is the Poisson bracket: $$\underbrace{[a(x,D), b(x,D)]}_{\text{order m_1+m_2-1}}\equiv\frac 1i\{a,b\}(x,D) \mod{\text{operators of order m_1+m_2-2.}}$$