Annals of Functional Analysis最新文献

Let (p(cdot ): {mathbb {R}}^nrightarrow (1,infty )) be a variable exponent, such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space (L^{p(cdot )}({mathbb {R}}^n),) and (phi : {mathbb {R}}^ntimes (0,infty )rightarrow (0,infty )) be a function satisfying some conditions. In this article, we give some properties of variable Campanato spaces ({mathcal {L}}_{p(cdot ),phi ,d}({mathbb {R}}^n),) with a non-negative integer d,  and variable Morrey spaces (L_{p(cdot ),phi }({mathbb {R}}^n),) and then establish their predual spaces. As an application of duality obtained in this article, we consider the boundedness of singular integral operators on variable Morrey spaces and their predual spaces.

New extensions of the Cauchy–Schwarz inequality in the framework of pre-Hilbert (C^*)-modules are given. An application to the numerical radius in (C^*)-algebras is also provided.

In this paper, we study rough Hausdorff operators on variable exponent Lebesgue spaces in the setting of the Heisenberg group. We prove the boundedness of rough Hausdorff operators by giving some sufficient conditions.

Given an open subset U of ({mathbb {C}}^n,) a weight v on U and a complex Banach space F,  let (mathcal {H}_v(U,F)) denote the Banach space of all weighted holomorphic mappings (f:Urightarrow F,) under the weighted supremum norm (left| fright| _v:=sup left{ v(z)left| f(z)right| :zin Uright} .) We prove that the set of all mappings (fin mathcal {H}_v(U,F)) that attain their weighted supremum norms is norm dense in (mathcal {H}_v(U,F),) provided that the closed unit ball of the little weighted holomorphic space (mathcal {H}_{v_0}(U,F)) is compact-open dense in the closed unit ball of (mathcal {H}_v(U,F).) We also prove a similar result for mappings (fin mathcal {H}_v(U,F)) such that vf has a relatively compact range.

If C and D are strictly accretive operators on ({mathcal {H}}) and at least one of them is normal, such that (CX!-!XDin { {{{varvec{{mathcal {C}}}}}}_{Psi }({mathcal {H}})}) for some (Xin { {{{varvec{{mathcal {B}}}}}}({mathcal H})}) and (Q^*) symmetrically norming function (Psi ,) then for all holomorphic functions h,  mapping the open right half (complex) plane into itself, we have (h( C,!)X!-!Xh(D)in { {{{varvec{{mathcal {C}}}}}}_{Psi }({mathcal {H}})},) satisfying

If (1leqslant q,r,sleqslant {+infty }) and (pgeqslant 2,A,B,Xin { {{{varvec{{mathcal {B}}}}}}({mathcal H})}) and A, B are strict contractions satisfying the condition (AX!-!XBin { {{{varvec{{mathcal {C}}}}}}_{s}({mathcal {H}})},) then for all holomorphic functions g,  mapping the open unit disc into the open right half (complex) plane, (g(A)X!-!Xg(B)in { {{{varvec{{mathcal {C}}}}}}_{s}({mathcal {H}})},) satisfying Schatten–von Neumann s-norms ((vert {;!vert {cdot }vert ;!}vert _s)) inequality

Other variants of some new Pick–Julia-type norm and operator inequalities are also obtained, they both complement the well-known Pick–Julia theorems for operators, obtained by Ky Fan, Ando, and Author, and they also extend these theorems to the field of norm ideals of compact operators, including Schatten–von Neumann ideals.

For (xin [0,1]), the run-length function (r_n(x)) is defined as the length of longest run of 1’s among the first n dyadic digits in the dyadic expansion of x. We study the Hausdorff dimension of the exceptional set in Erdös–Rényi limit theorem. Let (varphi :{{mathbb {N}}}rightarrow (0,infty )) be a monotonically increasing function with (lim _{nrightarrow infty }varphi (n)=infty ) and (0le alpha le beta le infty ), define

We prove that (E_{alpha ,beta }^varphi ) has Hausdorff dimension one if (lim _{n,prightarrow infty }frac{varphi (n+p)-varphi (n)}{p}=0) and that (E_{0,infty }^varphi ) is residual in [0,1] when (liminf _{nrightarrow {infty }}frac{varphi (n)}{n}=0).

Let Q, K be connected open subsets of (mathbb {R}^m) and A(X), A(Y) be some kind of function spaces. We will study the 2-local isometries between the vector-valued differentiable function spaces (C_0^p(Q, A(X))) and (C_0^p(K, A(Y))), and show that they can be written as weighted composition operators.

In this paper, we study the solution of the quadratic equation (TY^2-Y+I=0) where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series

where the sequence ((C_n)_{nge 0}) is the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator ((lambda T)^{-1}). Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers.