A common optimization problem is the minimization of a symmetric positive‎ ‎definite quadratic form $\langle x,Tx\rangle$ under linear‎ ‎constraints‎. ‎The solution to this problem may be given using the‎ ‎Moore-Penrose inverse matrix‎. ‎In this work at first we extend this‎ ‎result to infinite dimensional complex Hilbert spaces‎, ‎where a‎ ‎generalization is given for positive operators not necessarily‎ ‎invertible‎, ‎considering as constraint a singular operator‎. ‎A new‎ ‎approach is proposed when $T$ is positive semidefinite‎, ‎where the‎ ‎minimization is considered for all vectors belonging to‎ ‎$\mathcal{N}(T)^\perp$‎.

‎We obtain uniform boundedness principles for a new class of families‎ ‎of mappings from topological vector spaces to ordered topological‎ ‎vector spaces‎. ‎The new class of families of mappings includes the‎ ‎family of linear mappings and many other families which consist of‎ ‎nonlinear mappings‎.

‎If $\beta=\langle \beta_n\rangle_{n\in Z}$ is a sequence of positive numbers‎, ‎then a slant weighted Toeplitz‎ ‎operator $A_\phi$ is an operator on $L^2(\beta)$ defined as $A_\phi=WM_\phi$ where $M_\phi$ is the multiplication‎ ‎operator on $L^2(\beta)$‎. ‎When the sequence $\beta\equiv 1$‎, ‎this operator reduces to the ordinary slant Toeplitz operator‎ ‎given by M.C‎. ‎Ho in 1996‎. ‎In this paper‎, ‎we study some algebraic properties of the slant weighted Toeplitz operator‎. ‎We also obtain its matrix characterization and discuss the adjoint of this operator‎.

‎In this paper‎, ‎we obtain extensions of majorization type results‎ ‎and extensions of weighted Favard's and Berwald's inequality‎. ‎We‎ ‎prove positive semi-definiteness of matrices generated by‎ ‎differences deduced from majorization type results and differences‎ ‎deduced from weighted Favard's and Berwald's inequality‎. ‎This‎ ‎implies a surprising property of exponentially convexity and $\log$-convexity of these differences‎ ‎which allows us to deduce Lyapunov's and Dresher's inequalities for these‎ ‎differences‎, ‎which are improvements of majorization type results and‎ ‎weighted Favard's and Berwald's inequalities‎. ‎Analogous Cauchy's‎ ‎type means‎, ‎as equivalent forms of exponentially convexity and log-convexity‎, ‎are also studied‎ ‎and the monotonicity properties are proved‎.

‎Recently‎, ‎Suzuki [Nonlinear Anal‎. ‎71 (2009)‎, ‎no‎. ‎11‎, ‎5313-5317.]‎ ‎published a paper on which Edelstein's fixed theorem was generalized‎. ‎In this manuscript‎, ‎we give some theorems which‎ ‎are the generalization of the fixed theorem of Suzuki's Theorems and thus Edelstein's result [J‎. ‎London Math‎. ‎Soc‎. ‎37 (1962)‎, ‎74-79]‎.

‎In this paper we have established the fixed point theorem of self‎ ‎maps for $(\psi‎, ‎\phi)$-weak contractions in cone metric spaces‎. ‎Also our result is supported by an example‎.

We investigate a convex function $\psi_{p,q,\lambda}=\max \{\psi_p, \lambda \psi_q \}$, $(1\leq q\lt p\leq \infty)$, and its corresponding absolute normalized norm $\| .\|_{\psi_{p,q,\lambda}}$. We determine a dual norm and use it for getting refinements of the classical Hölder inequality. Also, we consider a related concave function $\phi_{p,q,\lambda}=\min \{\psi_p, \lambda \psi_q \}$, $(0\lt p\lt q\leq 1)$.

‎We apply a new version of multiscale convergence named very weak multiscale‎ ‎convergence to find possible frequencies of oscillation in an unknown‎ ‎coefficient of a partial differential equation from its solution‎. ‎We also‎ ‎use this notion to study homogenization of a certain linear parabolic‎ ‎problem with multiple spatial and temporal scales‎.

Recently, the class of $(\mathcal{C}_{p},\alpha)$-hyponormal operators is introduced and the Aluthge transforms of such operators is discussed by some researchers. This paper is to give a further development of the Aluthge transforms of $(\mathcal{C}_{p},\alpha)$-hyponormal operators by using Loewner-Heinz inequality, Furuta inequality and Lauric's lemma. Especially, it is shown that, if $p\ge 1$, $\alpha\ge 1/2$ and $T$ is $(\mathcal{C}_{p},\alpha)$-hyponormal, then the Aluthge transform $T(1/2,1/2)$ is $(\mathcal{C}_{4p\alpha/\beta},\beta)-hyponormal$ where $0 \lt \beta \le 1$ and $T(1/2,1/2)=|T|^{1/2}U|T|^{1/2}$.

‎An operator $T \in B(\mathcal{H}) $ is called quasi-class $(A‎, ‎k)$ if $T^{\ast‎ ‎k}(|T^{2}|-|T|^{2})T^{k} \geq 0$ for a positive integer $k$‎, ‎which‎ ‎is a common generalization of class A‎. ‎The famous Fuglede-Putnam's‎ ‎theorem is as follows‎: ‎the operator equation $AX=XB$ implies‎ ‎$A^{\ast}X=XB^{\ast}$ when $A$ and $B$ are normal operators‎. ‎In this‎ ‎paper‎, ‎firstly we show that if $X$ is a Hilbert-Schmidt operator‎, ‎$A$ is a quasi-class $(A‎, ‎k)$ operator and $B^{\ast}$ is an‎ ‎invertible class A operator such that $AX=XB$‎, ‎then‎ ‎$A^{\ast}X=XB^{\ast}$‎. ‎Secondly we consider the Putnam's inequality‎ ‎for quasi-class $(A‎, ‎k)$ operators and we also show that‎ ‎quasisimilar quasi-class $(A‎, ‎k)$ operators have equal spectrum and‎ ‎essential spectrum‎.

‎In this paper we prove a new discrete Hardy type inequality‎ ‎involving a kernel which has a more general form than those known‎ ‎in the literature‎. ‎We obtain necessary and sufficient conditions‎ ‎for the boundedness of a matrix operator from the weighted‎ ‎$l_{p,v}$ space into the weighted $l_{q‎, ‎u}$ space defined by‎ ‎$\left(Af\right)_j:=\sum\limits_{i=j}^\infty a_{i,j}f_i,$ for all‎ ‎$f=\{f_i\}_{i=1}^{\infty} \in l_{p,v}$ in case $q,p\in (1,\infty)$ with $q$ less than $p$, and‎ ‎$a_{i‎, ‎j}\geq 0$‎. ‎Then we deduce a corresponding dual statement‎.

‎In this paper we consider the ideal of Cohen $p$-nuclear multilinear‎ ‎mappings‎, ‎which is a natural multilinear extension of the ideal of‎ ‎$p$-nuclear linear operators‎. ‎The space of Cohen $p$-nuclear‎ ‎$m$-linear mappings is characterized by means of a suitable tensor‎ ‎norm up to an isometric isomorphism‎.

‎In this survey we present some recent results obtained by the author in‎ ‎extending Ostrowski inequality in various directions for continuous‎ ‎functions of selfadjoint operators defined on complex Hilbert spaces‎.

‎In this paper we consider the question when an upward directed set of positive ideal-triangularizable‎ ‎operators on a Banach lattice is (simultaneously) ideal-triangularizable‎. ‎We prove that a majorized upward directed set of ideal-triangularizable positive operators‎, ‎which are compact or abstract integral operators is ideal-triangularizable‎. ‎We also prove that a finite subset of an additive semigroup of positive power compact quasinilpotent operators is ideal-triangularizable‎. ‎Moreover‎, ‎we prove that an additive semigroup of positive power compact quasinilpotent operators of bounded compactness index is ideal-triangularizable‎.