An operator $T$ on the separable infinite-dimensional Hilbert space is constructed so that the commutant of every operator which is not a scalar multiple of the identity operator and commutes with $T$ coincides with the commutant of $T$. On the other hand, it is shown that for several classes of operators it is possible to construct a finite sequence of operators, starting at a given operator from the class and ending in a rank-one projection such that each operator in the sequence commutes with its predecessor. The classes which we study are: finite-rank operators, normal operators, partial isometries, and $C_0$ contractions. It is also shown that for any given set of yes/no conditions between points in some finite set, there always exist operators on a finite-dimensional Hilbert space such that their commutativity relations exactly satisfy those conditions.
COBISS.SI-ID: 16556377
This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free $\ast$-algebras $\mathbb{R} \langle x, x^\ast \rangle$ with $x = (x_1, \dots , x_n)$. First, we introduce notions of the (noncommutative) zero set of a left ideal and of a real left ideal. We prove that every element from $\mathbb{R} \langle x, x^\ast \rangle$ whose zero set contains the intersection of zero sets of elements from a finite subset $S$ of $\mathbb{R} \langle x, x^\ast \rangle$ belongs to the smallest real left ideal containing $S$. Next, we give an implementable algorithm, which for every finite $S \subset \mathbb{R} \langle x, x^\ast \rangle$, computes the smallest real left ideal containing $S$, and prove that the algorithm succeeds in a finite number of steps. Our definitions and some of our results also work for other $\ast$-algebras. As an example, we treat real left ideals in $M_n(\mathbb{R}[x_1])$.
COBISS.SI-ID: 16636249
Given linear matrix inequalities (LMIs) $L_1$ and $L_2$ it is natural to ask: $(Q_1)$ when does one dominate the other, that is, does $L_1(X) \succeq 0$ imply $L_2(X) \succeq 0$? $(Q_2)$ when are they mutually dominant, that is, when do they have the same solution set? The matrix cube problem of Ben-Tal and Nemirovski (SIAM J Optim 12:811-833, 2002) is an example of LMI domination. Hence such problems can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables $x_j$. With this relaxation, the domination questions $(Q_1)$ and $(Q_2)$ have elegant answers, indeed reduce to constructible semidefinite programs. As an example, to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in Ben-Tal and Nemirovski (SIAM J Optim 12:811-833, 2002). Thus our relaxation could be viewed as generalizing it. Assume there is an $X$ such that $L_1(X)$ and $L_2(X)$ are both positive definite, and suppose the positivity domain of $L_1$ is bounded. For our "matrix variable" relaxation a positive answer to $(Q_1)$ is equivalent to the existence of matrices $V_j$ such that $$L_2(x) = V_1^\ast L_1 (x)V_1 + \cdots + V_\mu^\ast L_1(x) V_\mu. \qquad \qquad (A_1)$$ As for $(Q_2)$ we show that $L_1$ and $L_2$ are mutually dominant if and only if, up to certain redundancies described in the paper, $L_1$ and $L_2$ are unitarily equivalent. Algebraic certificates for positivity, such as $(A_1)$ for linear polynomials, are typically called Positivstellensätze. The paper goes on to derive a Putinar-type Positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form $\{X|L(X) \succeq 0\}$. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map $\tau$ from a subspace of matrices to a matrix algebra is "completely positive". Complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras. On one hand it provides tools for studying LMIs and on the other hand, since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts, the theory of completely positive maps provides perspective on the difficulties in solving LMI domination problems.
COBISS.SI-ID: 16592985
The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f , what is the smallest trace f (A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix -algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other sidetwo topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.
COBISS.SI-ID: 2048170515
This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several examples are given illustrating the results.
COBISS.SI-ID: 2048184851