The paper studies finite extensions of Bessel sequences in infinite-dimensional Hilbert spaces. We provide a characterization of Bessel sequences that can be extended to frames by adding finitely many vectors. We also characterize frames that can be converted to Parseval frames by finite-dimensional perturbations. Finally, some results on excesses of frames and near-Riesz bases are derived.

Let ${\Phi} =(\phi_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{{\Phi}}^{A}(X)$ is introduced. It is shown that the space is a complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{{\Phi}}^{A}(X)$ is a $\sigma$-Dedekind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.

Let $X$ be a pointed metric space and let $E$ be a Banach space. It is known that the Lipschitz space $\mathrm{Lip}_o(X,E^*)$ is isometrically isomorphic to $(\mathcal{F}(X)\widehat{\otimes}_\pi E)^*$, the dual of the projective tensor product of the Lipschitz-free space $\mathcal{F}(X)$ and $E$. Since the injective norm $\varepsilon$ on $\mathcal{F}(X)\otimes E$ is smaller than the projective norm $\pi$, we study Lipschitz Grothendieck-integral operators which are exactly those elements in $\mathrm{Lip}_o(X,E^*)$ which correspond to the elements of $(\mathcal{F}(X)\widehat{\otimes}_\varepsilon E)^*$, the dual of the injective tensor product of $\mathcal{F}(X)$ and $E$.

The main purpose of this article is to develop the generalized analytic Fourier--Feynman transform theory. We introduce a generalized analytic Fourier--Feynman transform and a multiple generalized analytic Fourier--Feynman transform with respect to Gaussian processes on the function space $C_{a,b}[0,T]$ induced by a generalized Brownian motion process. We then establish a relationship between these two generalized analytic transforms.

In Hilbert spaces, the redundancy property of $g$-frames is different from that of frames, and the dilation theory is interesting and important in many mathematical fields. In this paper, we study the redundancy and dilations of $g$-frames in Hilbert spaces. First, we characterize $g$-Riesz bases and exact $g$-frames under some constraints, then obtain some dilation results for (normalized tight) $g$-frames, and give some properties about them. Finally we prove some interesting properties on the canonical duals of $g$-frames.

Utilizing the gap between homogenous subsets which is introduced in this paper, the perturbations for the Moore--Penrose metric generalized inverses of bounded linear operators in Banach spaces are discussed. Under range--preseving, kernel--preseving and general case, respectively, we get some new results about error estimate of the perturbations for the Moore--Penrose metric generalized inverse of bounded linear operators.

We study the noncommutative Orlicz modular spaces associated with growth functions. Some basic properties of such spaces, such as completeness and dominated convergence theorem, are present. Moreover, Young inequalities and Clarkson--McCarthy inequalities on these spaces proved.

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.

In this article, the authors first introduce the Triebel--Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ with variable exponents, and establish its $\varphi$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ via Peetre maximal functions.

We study the diameter two properties in the (James type) Banach spaces $JH$, $JT_\infty$ and $JH_\infty$. We show that the dual spaces of these three Banach spaces fail every diameter two property. Also, we prove that $JH$ and $JH_{\infty}$ satisfy the strong diameter two property, and so the dual norms of these spaces are octahedral. In addition, we find a closed hyperplane $M$ of $JH_\infty$ such that its dual space, $M^*$, satisfies the $w^*$-strong diameter two property. Finally, we get that the natural norms of $M$ and $M^*$ are octahedral.

We study algebraic and topological properties of the group of all surjective isometries on several spaces of vector valued analytic functions and vector valued $L^p$ spaces ($1 \leq p \leq \infty$). We also derive the form for the surjective linear isometries on the vector valued little Zygmund space.

The concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity. We then define a type of open (closed) sets called open (closed) unit neighborhoods of $0$. We show among other things that in $\mathbb{R} $ and $\mathbb{C} $ the only non-trivial open and closed unit neighborhoods of $0$ are the open unit ball and the closed unit ball, respectively.

We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible Bishop--Phelps--Bollobás version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of $w^*$-continuous operators and other related results.

Let $\Gamma$ be a fractal $h$-set and ${\mathbb{B}}^{\sigma}_{p,q}(\Gamma)$ be a trace space of Besov type defined on $\Gamma$. While we dealt in our earlier papers with growth envelopes of such spaces mainly and investigated the existence of traces in detail, we now study continuous embeddings between different spaces of that type on $\Gamma$. We obtain necessary and sufficient conditions for such an embedding to hold, and can prove in some cases complete characterisations. It also includes the situation when the target space is of type $L_r(\Gamma)$ and, as a by-product, under mild assumptions on the $h$-set $\Gamma$ we obtain the exact conditions on $\sigma$, $p$ and $q$ for which the trace space ${\mathbb{B}}^{\sigma}_{p,q}(\Gamma)$ exists. We can also refine some embedding results for spaces of generalised smoothness on $\mathbb{R}^n$.

Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollobás proved the nowadays known as the Bishop--Phelps--Bollobás theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop--Phelps--Bollobás moduli of a Banach space have been introduced $[\textit{J. Math. Anal. Appl.}~412 (2014), 697--719]$ to measure, for a given Banach space, what is the best possible Bishop--Phelps--Bollobás theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop--Phelps--Bollobás modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop--Phelps--Bollobás modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop--Phelps--Bollobás modulus.

Consider a closed $O^*$--algebra $\mathcal{M}$ on a dense linear subspace $\mathcal{D}$ of a Hilbert space $\mathcal{H}$, a locally compact group $G$ with left invariant Haar measure $ds$ and an action $\alpha$ of $G$ on $\mathcal{M}$. Under some natural conditions, the $O^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ of $\mathcal{M}$ and the $GW^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ are introduced. When $G$ is also abelian, the dual action $\widehat{\alpha}$ of the dual group $\widehat{G}$ on $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ and on $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ is defined, which makes it possible to study the crossed products $(\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{O^*}{\rtimes}}\widehat{G}$ and $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{GW^*}{\rtimes}}\widehat{G}$. In case of modular actions, these constructions are used to obtain results on duality of type $\mathrm{II}$--like and type $\mathrm{III}$--like $GW^*$--algebras.

In this paper, we introduce a certain Herz-type Hardy space with variable exponent and establish the atom decomposition theorem for it. Using this decomposition, we obtain some boundedness on the Herz-type Hardy space with variable exponent for a class of pseudo-differential operators.

We investigate subalgebras $A$ of the Cuntz algebra ${\mathcal O}_n$ that arise as finite direct sums of corners of the UHF-subalgebra ${\mathcal F}_n$. For such an $A$, we completely determine its normalizer group inside ${\mathcal O}_n$.

Matrix splitting and its convergence theorems are useful tools for finding solution of linear system of equations, iteratively. In this article, we introduce a few matrix splittings arising from index-proper splittings. Then their convergence results and their applications to multisplitting theory are studied.

We treat several questions related to the positions of subspaces of $\ell_1$. Among them, we show that all quotients $\ell_1/\ell_1$ have the Schur property and that a nontrivial twisted sum of $\ell_1$ and $c_0$ cannot be isomorphic to the direct product $\ell_1 \oplus c_0$.