In this paper, we show Hilbert $C^*$-module versions of Hölder--McCarthy inequality and its complementary inequality. As an application, we obtain Hölder type inequalities and its reverses on a Hilbert $C^*$-module.

In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework.

We prove that if positive invertible operators $A$ and $B$ satisfy an operator inequality $(B^{s/2}A^{(s-t)/2}B^tA^{(s-t)/2}B^{s/2})^{1\over{2s}}\geq B$ for some $t > s > 0$, then

(1) If $t \ge 3s-2 \ge 0$, then $\log B \geq \log A$, and if $t \ge s+2$ is additionally assumed, then $B \ge A$.

(2) If $s \in (0, 1/2)$, then $\log B \geq \log A$, and if $t \ge s+2$ is additionally assumed, then $B \ge A$.

It is an interesting application of the Furuta inequality. Furthermore we consider some related results.

The zero-dilation index $d(A)$ of an $n$-by-$n$ complex matrix $A$ is the maximum size of the zero matrix which can be dilated to $A$. In this paper, we determine the value of this index for a certain KMS matrix by using the Li--Sze characterization of higher-rank numerical ranges of a finite matrix.

A criterion for the numerical range of a linear operator acting in a Krein space to be a two-component hyperbolical disc is given, using the concept of support function. A characterization of the Krein space numerical range as a union of hyperbolical discs is obtained by a reduction to the two-dimensional case. We revisit a famous result of Ando concerning the inclusion relation $W(A)\subseteq W(B)$ of the numerical ranges of two operators $A$ and $B$ acting in (possibly different) Hilbert spaces, and the condition that $A$ can be dilated to an operator of the form $B\otimes I$. The extension of this result to operators acting in Krein spaces is investigated.

For semisimple, commutative Banach algebras $\mathcal A$ and $\mathcal B$, some Banach algebra properties of the Cartesin product $\mathcal A \times \mathcal B$ are characterized in terms of $\mathcal A$ and $\mathcal B$. A couple of results are also proved for non-commutative Banach algebras.

We apply numerical radius and spectral radius estimates to the Frobenius companion matrices of monic polynomials to derive new bounds for their zeros and give different proofs of some known bounds.

We investigate $2$-local$^*$-automorphisms, $2$-local$^*$-antiautomorphisms, and $2$-local Jordan$^*$-derivations on certain algebras with involution.

In the present paper, by Haagerup theorem, we show that if $A \in \mathbb{M}_{n}$ is a non scalar strictly positive matrix and $\nu \in (0,1)$ be a real number such that $ \nu \neq \frac{1}{2},$ then there exists $X \in \mathbb{M}_{n}$ such that $$\|A^{\nu}XA^{1-\nu}\| > \| \nu AX + (1- \nu)XA\|.$$

In this paper several new inequalities of Grüss' type for Riemann--Stieltjes integral with monotonic nondecreasing integrators under various assumptions for integrands are proved. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

We give a complete relation between the singular values and eigenvalues of a complex skew symmetric matrix in terms of multiplicative majorization and double occurrences of singular values and eigenvalues. Similar studies are given for matrices in the algebras $\mathfrak{sp}_n(\mathbb{C})$ and $\mathfrak{sp}_n(\mathbb{R})$.

Let $a, b, c$ and $d$ be functions in $L^2 = L^2(\mathbb{T}, d\theta/2\pi)$, where $\mathbb{T}$ denotes the unit circle. Let $\mathcal{P}$ denote the set of all trigonometric polynomials. Suppose the singular integral operators $A$ and $B$ are defined by $A=aP+bQ$ and $B = cP+dQ$ on $\mathcal{P}$, where $P$ is an analytic projection and $Q = I-P$ is a co-analytic projection. In this paper, we use the Helson--Szegő type set $(HS)(r)$ to establish the condition of $a, b, c$ and $d$ satisfying $\|Af\|_2 \geq \|Bf\|_2$ for all $f$ in $\mathcal{P}$. If $a, b, c$ and $d$ are bounded measurable functions, then $A$ and $B$ are bounded operators, and this is equivalent to that $B$ is majorized by $A$ on $L^2$, i.e., $A^*A \geq B^*B$ on $L^2$. Applications are then presented for the majorization of singular integral operators on weighted $L^2$ spaces, and for the normal singular integral operators $aP + bQ$ on $L^2$ when $a-b$ is a complex constant.

In this paper we introduce a strong version of the Birkhoff--James orthogonality in Hilbert $C^*$-modules. More precisely, we consider elements $x$ and $y$ of a Hilbert $C^*$-module $V$ over a $C^*$-algebra $A$ which satisfy $\|x\|\le \|x+ya\|$ for all $a\in A.$ We show that this relation can be described as the Birkhoff--James orthogonality of appropriate elements of $V,$ and characterized in terms of states acting on the underlying $C^*$-algebra $A.$ Some analogous relations of this type are considered as well.

In this paper we study positive operator monotone functions on $(0, 1)$ which have some differences from those on $(0, \infty):$ we show that for a concave operator monotone function $f$ on $(0, 1),$ the Kwong matrices $K_f(s_1, \dots, s_n)$ are positive semidefinite for all $n$ and $s_i \in (0, 1),$ and $f(s^p)^{1/p}$ for $p \in (0,1]$ and $s/f(s)$ are operator monotone. We also give a sufficient condition for the Kwong matrices to be positive semidefinite.

Let $H$ be a complex Hilbert space. We study the geometry of the space of pairs $(A, E)$, for $A$ a (semidefinite bounded linear) positive operator on $H$ and $E$ a (bounded linear) projection on $H$ such that $AE=E^*A.$