### Bilipschitz mappings and strong ${A}_{\infty}$ weights.

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The study of circumcenters in different types of triangles in real normed spaces gives new characterizations of inner product spaces.

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathcal{B}\left(\mathscr{H}\right)$ and ${\mathcal{B}}_{A}\left(\mathscr{H}\right)$ the sub-algebra of $\mathcal{B}\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{\mathcal{B}}_{A}\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi $ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi $ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left({T}^{\u266f}\right)=\psi {\left(T\right)}^{\u266f}$ for all $T\in {\mathcal{B}}_{A}\left(\mathscr{H}\right)$, where $P={A}^{+}A$ and ${A}^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi $ preserves $A$-quasi-unitary operators in both directions such that there exists an...

This paper is a continuation of investigations of $n$-inner product spaces given in [five, six, seven] and an extension of results given in [three] to arbitrary natural $n$. It concerns families of projections of a given linear space $L$ onto its $n$-dimensional subspaces and shows that between these families and $n$-inner products there exist interesting close relations.

Let $E$ be a real linear space. A vectorial inner product is a mapping from $E\times E$ into a real ordered vector space $Y$ with the properties of a usual inner product. Here we consider $Y$ to be a $\mathcal{B}$-regular Yosida space, that is a Dedekind complete Yosida space such that ${\bigcap}_{J\in \mathcal{B}}J=\left\{0\right\}$, where $\mathcal{B}$ is the set of all hypermaximal bands in $Y$. In Theorem 2.1.1 we assert that any $\mathcal{B}$-regular Yosida space is Riesz isomorphic to the space $B\left(A\right)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval...

Some generalized notions of James' orthogonality and orthogonality in the Pythagorean sense are defined and studied in the case of generalized normed spaces derived from generalized inner products.

We consider positive definite kernels which are invariant under a multiplier and an action of a semigroup with involution, and construct the associated projective isometric representation on a Hilbert C*-module. We introduce the notion of C*-valued Hilbert-Schmidt kernels associated with two sequences and construct the corresponding reproducing Hilbert C*-module. We also discuss projective invariance of Hilbert-Schmidt kernels. We prove that the range of a convolution type operator associated with...

In this paper we introduce two mappings associated with the lower and upper semi-inner product ${(\xb7,\xb7)}_{i}$ and ${(\xb7,\xb7)}_{s}$ and with semi-inner products $[\xb7,\xb7]$ (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.