Abstract:
We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we show the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X,\mu)$ with the spectral radius $r(K)$. Then the function $\phi$ defined by $\phi(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, {1/2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X,\mu)$.

Abstract:
Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997), 443-456] and [G. Cigler, R. Drnov\v{s}ek, D. Kokol-Bukov\v{s}ek, T. Laffey, M. Omladi\v{c}, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998), 452-465].

Abstract:
K. He, J. Hou, and M. Li have recently given a sufficient and necessary condition for unitary equivalence of quantum states. This condition is based on the von Neumann entropy. In this note we first give a short proof of their result, and then we improve it.

Abstract:
Let A and B be bounded operators on a Banach lattice E such that the commutator C=AB-BA and the product BA are positive operators. If the product AB is a power-compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. This theorem answers an open question posed in a paper by Bra\v{c}i\v{c}, Drnov\v{s}ek, Farforovskaya, Rabkin and Zem\'{a}nek.

Abstract:
Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0, 1]. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. His proof is based on the Kiefer-Wolfowitz equivalence theorem from the approximate theory of optimal design. In this note we give a short and simple proof of his result.

Abstract:
Let $A = [a_{i j}]_{i,j=1}^n$ be a nonnegative matrix with $a_{1 1} = 0$. We prove some lower bounds for the spread $s(A)$ of $A$ that is defined as the maximum distance between any two eigenvalues of $A$. If $A$ has only two distinct eigenvalues, then $s(A) \ge \frac{n}{2(n-1)} \, r(A)$, where $r(A)$ is the spectral radius of $A$. Moreover, this lower bound is the best possible.

Abstract:
For any measurable set $E$ of a measure space $(X, \mu)$, let $P_E$ be the (orthogonal) projection on the Hilbert space $L^2(X, \mu)$ with the range $ran \, P_E = \{f \in L^2(X, \mu) : f = 0 \ \ a.e. \ on \ E^c\}$ that is called a standard subspace of $L^2(X, \mu)$. Let $T$ be an operator on $L^2(X, \mu)$ having increasing spectrum relative to standard compressions, that is, for any measurable sets $E$ and $F$ with $E \subseteq F$, the spectrum of the operator $P_E T|_{ran \, P_E}$ is contained in the spectrum of the operator $P_F T|_{ran \, P_F}$. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator $T$ has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space $(X, \mu)$ is discrete or the operator $T$ has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.

Abstract:
This empirical study describes the functions of visual acts of meaning in the educational communication between teachers and their pupils. The study understands visual acts to be visual messages that have meaning for participants of communication. The data was gathered through direct observation of 32 classes of humanities subjects at lower secondary schools. Visual acts were firstly divided into topic visual acts (which convey events, objects or thoughts directly related to the topic of conversation) and interactive visual acts (which can be characterized as contact messages). In the context of lessons, topic visual acts are presented as a tool for conceptualisation of educational content. On the other side, interactive visual acts are described as a mechanism which participates in establishing the setting of educational communication. Although visual acts of meaning often have the same functions and structure, it transpires that their nature is highly individualized and it can be seen as a sign of a teacher’s professional identity.