Abstract Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on Hilbert space $\mathcal{H}$ . The multiplicity $T$ is the cardinality minimal generating set with respect to In this paper, we establish an additive formula for multiplicities class tuples operators. A special case main result states following: $n \geq 2$ and let $\mathcal{Q}_i$ $i 1, n$ proper closed shi...

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

In the present article, we investigate a possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor’s K-group of the real numbers into the additive group of real numbers. Usi...

We extend the notion of ascent and descent for an operator acting on a vector space to sets of operators. If the ascent and descent of a set are both finite then they must be equal and give rise to a canonical decomposition of the space. Algebras of operators, unions of sets and closures of sets are treated. As an application we construct a Browder joint spectrum for commuting tuples of bounded...

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of co...