Narrow Results

The self-dual cone — the central object of this review — is introduced. Several operator inequalities associated with the self-dual cone are defined and mathematical properties of those are investigated. In general there are infinitely many choices of self-dual cones in a Hilbert space. Each of these lead to a distinct family of operator inequalities in the Hilbert space which enables us to analyze quantum physical models with respect to several aspects. We refer to these applications as self-dual cone analysis. The focus of this review lies on the self-dual cone analysis of models in condensed matter physics. In particular, by taking a physically proper self-dual cone, the interaction term of the Hamiltonian of the system becomes attractive from a viewpoint of our new operator inequalities. This attractive term enables us to analyze the system and various aspects of physical interest in detail. For instance, if the attractive term is ergodic, it is shown that the ground state is unique. By the uniqueness and the conservation laws, the physically symmetric state is realized as the ground state. This could be regarded as a physical order. As applications, the BCS model and the one-dimensional Fröhlich model are studied. We explain, from a viewpoint of the self-dual cone analysis, the appearance of macroscopic phase angles in the superconductors, Josephson effect and the Peierls instability.

We investigate Schrödinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrödinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ′-interactions.

In this paper, using the contractive maps, we present some new interpolations between the Heinz and Heron operator means for unitarily invariant norms. Let $A,X,B\in B(H)$ and $A,B$ be two positive definite operators. and let $\frac{1}{2}\le \beta \le 1$, $\alpha \in [\frac{1}{2},\infty )$. If $\frac{1}{4}\le \nu \le \mu \le \frac{1}{2}$ and $\frac{\mu +\nu }{2}\le r\le 1-\frac{\mu +\nu }{2}$, or if $\frac{1}{2}\le \mu \le \nu \le \frac{3}{4}$ and $1-\frac{\mu +\nu }{2}\le r\le \frac{\mu +\nu }{2}$, then

In this paper, we present a refinement of the well-known arithmetic-geometric mean inequality. As application of our result, we obtain an operator inequality. We give an improvement of the inequality presented by Kittaneh for the numerical radius.

In this paper we propose some Heinz-type inequalities for convex functions which provide a very simple proof of the main results of Kittaneh, Moslehian and Sababheh showed in the paper (F. Kittaneh, M. S. Moslehian and M. Sababheh, Quadratic interpolation of the Heinz means, Math. Inequal. Appl. 21(3) (2018) 739–757). We apply these inequalities to infer new inequalities for power means. As an application, we also give operator versions of Heinz-type inequalities for power and Heinz means.