Lower bounds for norms of commutators

Question

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?

Answer 1

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.

Answer 2

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: $$ \underbrace{[a(x,D), b(x,D)]}_{\text{order $m_1+m_2-1$}}\equiv\frac 1i\{a,b\}(x,D) \mod{\text{operators of order $m_1+m_2-2$.}} $$