We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

We completely determine the ${\ell }_q$ and C(K) spaces which are isomorphic to a subspace of ${\ell }_p\otimes {̂}_{\pi }C\left(\alpha \right)$, the projective tensor product of the classical ${\ell }_p$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from ${\ell }_p$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states...

Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G=G_p\times B$ is a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $S^{\lambda }G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $S^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $S^{\lambda }{G}_p$-module V and an irreducible...

Let $\mathbb{Z}{̂}_p$ be the ring of p-adic integers, $U\left(\mathbb{Z}{̂}_p\right)$ the unit group of $\mathbb{Z}{̂}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $\mathbb{Z}{̂}_p^{\lambda }G$ the twisted group algebra of G over $\mathbb{Z}{̂}_p$ with a 2-cocycle $\lambda \in Z²(G,U\left(\mathbb{Z}{̂}_p\right))$. We give necessary and sufficient conditions for $\mathbb{Z}{̂}_p^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $\mathbb{Z}{̂}_p^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $\mathbb{Z}{̂}_p^{\lambda }{G}_p$-module V and an irreducible $\mathbb{Z}{̂}_p^{\lambda }B$-module W.

Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

A Banach space $X$ has the reciprocal Dunford-Pettis property ($RDPP$) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $RDPP$ (resp. property $\left(wL\right)$) if every $L$-subset of $X^{*}$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X\otimes {}_{\pi }Y$ has property $\left(wL\right)$ when $X$ has the $RDPP$, $Y$ has property $\left(wL\right)$, and $L(X,Y^{*})=K(X,Y^{*})$.

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

It is shown that the Bourgain algebra $A{\left(\right)}_b$ of the disk algebra A() with respect to $H^{\infty }\left(\right)$ is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to $H^{\infty }\left(\right)$, the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is $A{\left(\right)}_b$.

For the groups $GU\left(3\right)$, $SU\left(3\right)$, $GL\left(3\right)$, $SU\left(3\right)$ over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between...