Sharp bounds on the ranks of negativity of certain sums

Abstract

If M is a complex vector space and 〈·, ·〉 a Hermitian sesquilinear form on M with a finite rank of negativity k (i.e., k is the maximal dimension of any linear subspace E of M satisfying 〈x, x〉 < 0 for each nonzero x in E), if n is a positive integer, and if a ₁, …, a _n are endomorphisms of M, then it is easy to see that the Hermitian sesquilinear form

$$ (x,y) \mapsto \sum\limits_{v = 1}^n {\left\langle {a_v x,a_v y} \right\rangle } $$

on M has rank of negativity at most nk. It is also fairly easy to see that the bound nk cannot be improved in general. Less trivial is the fact that it cannot be improved by making the following assumption

(a) the space M is the *-algebra A:= (C[[w ₁, w ₂]] of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form φ on A such that 〈x, y〉 = φ(y* x) (x, y ∈ A); and a _v is just left multiplication by some element of A (which we may denote by ‘a _v ’ at no great risk of confusion).

Now suppose that, with M, 〈·, ·〉, k, n, and a ₁ , …, a _n as initially, the following two conditions are satisfied:

• (i) each a _v has a formal adjoint a* _v , being an endomorphism of M such that

  $$ \left\langle {a_v x,y} \right\rangle = \left\langle {x,a_v^* y} \right\rangle (x,y \in M); $$

• (ii) the mappings a ₁, …, a _n , a*₁, …, a* _n commute pairwise.

Then the bound nk can be replaced by k (regardless of how large n may be). This result cannot be improved in general since it may happen that each a _v is a scalar multiple of the identical mapping of M into itself (not all a _v equal to 0), in which case the form (1) is a positive multiple of 〈·, ·〉 itself.

There are ties with the subjects of ‘positive semidefinite submodules’ (‘positive semidefinite left ideals’) and ‘definitisation’.