Related papers: On the operator equation AX-XB=C

Based on the conjecture for the exact eigenvalue of the transfer matrix of the higher half-integer spin XXZ chain at the Razumov-Stroganov point, we evaluate the corresponding Baxter's Q operator in closed form by solving the TQ equation.…

Rota-Baxter operators were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. The polynomial algebra $\mathbf{k}[x]$ plays a central role both in analysis…

Let $f \geq 0$ be operator monotone on $[0, \infty)$. In this paper we prove that for any unitarily-invariant norm $|||-|||$ on $M_n(\mathbb{C})$ and matrices $A, B, X \in M_n(\mathbb{C})$ with $A, B \geq 0$ and $|||X||| \leq 1$,…

This paper provides a view of Maxwell's equations from the perspective of complex variables. The study is made through complex differential forms and the Hodge star operator in $\mathbb{C}^2$ with respect to the Euclidean and the Minkowski…

Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear bounded operator. It was proved by C. Foias and J.P. Williams that in certain cases the operator block matrix $R(X;T,V)$ (defined in the text) is similar to a…

Let A be a linear space of operators on a Hilbert space H, x a vector in H, and Ax the subspace of H comprising all vectors of the form Tx with T in A. We discuss, within a Bishop-style constructive framework, conditions under which the…

The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert…

A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX - C)^{#}(BX -C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a…

A Markov operator $P$ acting on $C(X)$, where $X$ is compact, gives rise to a natural topological quiver. We use the theory of such quivers to attach a $C^{*}$-algebra to $P$ in a fashion that reflects some of the probabilistic properties…

We solve the Yang-Baxter-like matrix equation $AXA = XAX$ for a general given matrix $A$ to get all anti-commuting solutions, by using the Jordan canonical form of $A$ and applying some new facts on a general homogeneous Sylvester equation.…

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient…

We present a new approach to the question of when the commutativity of operator exponentials implies that of the operators. This is proved in the setting of bounded normal operators on a complex Hilbert space. The proofs are based on some…

The real theory of the Dunkl operators has been developed very extensively, while there still lacks the corresponding complex theory. In this paper we introduce the complex Dunkl operators for certain Coxeter groups. These complex Dunkl…

ZX-calculus is a strict mathematical formalism for graphical quantum computing which is based on the field of complex numbers. In this paper, we extend its power by generalising ZX-calculus to such an extent that it is universal both in an…

Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in…

We introduce a $Q$-operator $\mathcal{Q}_z$ for the hyperbolic Calogero--Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a $Q$-operator, i.e.~invariance of Hamiltonians,…

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator…

Let $E \ni x\mapsto A(x)$ be a $\mathscr{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr{C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$…

We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to…

In this master thesis, I discuss how the theory of operator algebras, also called operator theory, can be applied in quantum computer science.