On decay rates of the solutions of parabolic Cauchy problems

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝ N . We show that given a weighted L p -space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists n m > 0 such that, if the initial data f belongs to the closed linear span of e n with n ⩾ n m , then the decay rate of the solution of the problem is at least t−m for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.

Character Density in Central Subalgebras of Compact Quantum Groups

We investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on L2(𝔾) and use this result to show the weak* density and normdensity of characters in ZL∞(G) and ZC(G), respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of L1(G), we show that the center Z(L1(G)) is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that Z(L1(G)) is a completely complemented Z(L1(G))-submodule of L1(G).

Linear rigidity of stationary stochastic processes

We consider stationary stochastic processes $\{X_{n}:n\in \mathbb{Z}\}$ such that $X_{0}$ lies in the closed linear span of $\{X_{n}:n\neq 0\}$; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class $\unicode[STIX]{x1D6EC}_{\ast }(1)$. We next give a sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be linearly rigid. Finally, we show that the determinantal point process on $\mathbb{R}^{2}$ induced by a tensor square of Dyson sine kernels is not linearly rigid.

A note on bump functions that locally depend on finitely many coordinates

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

1-Complemented Subspaces of Spaces With 1-Unconditional Bases

We prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ22 and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓp(ℓq) for 1 ≤ p,q > ∞.Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓp for some 1 ≤ p,q > ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Some results on the span of families of Banach valued independent, random variables

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

RUC-systems and Besselian systems in Banach spaces

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the seriesconverges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such thatfor each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

Approximation on Boundary Sets

Let U be a bounded open subset of the complex plane. By a well known result of A. M. Davie, C(bU) is the uniformly-closed linear span of A(U) and the powers (z-zi)-n, n = 1, 2, 3, … with zi a point in each component of U. We show that if A(U) is a Dirichlet algebra and bU is of infinite length, then one power of (z - zi) is superfluous.

A Note on Unconditional Bases

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that1.1convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by1.2where then fi(xj) = δij and fi∊X* for each positive integer i.A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).