 A chain condition for operators from \(C(K)\)spaces
przyjęta do druku w Quarterly Journal of Mathematics
Abstrakt
We introduce a chain condition (
), defined for operators acting on \(C(K)\!\)spaces, which is intermediate between weak compactness and having a weakly compactly generated range. It is motivated by Pełczyński's characterisation of weakly compact operators on \(C(K)\!\)spaces. We prove that if \(K\) is extremally disconnected and \(X\) is a Banach space then, for an operator \(T\colon C(K)\to X,\) \(T\) is weakly compact if and only if \(T\) satisfies (
) if and only if the representing vector measure of \(T\) satisfies an analogous chain condition. As a tool for proving the abovementioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces \(K\) for which the identity operator on \(C(K)\) satisfies (
), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramseytype theorem, due to Dushnik and Miller, we prove that the collection of operators on a \(C(K)\!\)space satisfying (
) forms a closed left ideal of \(\mathscr{B}(C(K))\).
 Maximal left ideals of the Banach algebra of bounded operators on a Banach space
przyjęta do druku w Studia Mathematica
Abstrakt
We address the following two questions regarding the maximal left ideals of the Banach algebra \(\mathscr{B}(E)\) of bounded operators acting on an infinitedimensional Banach space \(E\):
(I) Does \(\mathscr{B}(E)\) always contain a maximal left ideal which is not finitely generated?
(II) Is every finitelygenerated, maximal left ideal of \(\mathscr{B}(E)\) necessarily of the form $$(\ast)\qquad\{T\in\mathcal{B}(E)\colon\, Tx=0\}\quad$$ for some nonzero \(x\in E\)?
Since the twosided ideal \(\mathscr{F}(E)\) of finiterank operators is not contained in any of the maximal left ideals given by (\(\ast\)), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinitedimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitelygenerated, maximal left ideal of \(\mathscr{B}(E)\) contains \(\mathscr{F}(E)\); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.
 The ideal of weakly compactly generated operators acting on a Banach space
przyjęta do druku w Journal of Operator Theory
Abstrakt
We call a bounded linear operator acting between Banach spaces weakly compactly generated (\(\mathsf{WCG}\) for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of \(\mathsf{WCG}\) operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the \(p\)th long James space \(\mathcal{J}_p(\omega_1)\), we show how properties of the ideal of \(\mathsf{WCG}\) operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the \(K_0\)group of \(\mathscr{B}(\mathcal{J}_p(\omega_1))\) as the additive group of integers.
 Stability of vector measures and twisted sums of Banach spaces
Journal of Functional Analysis 264 (2013), 24162456.
Abstrakt
A Banach space \(X\) is said to have the \(\mathsf{SVM}\) (stability of vector measures) property if there exists a constant \(v<\infty\) such that for any algebra of sets \(\mathscr{F}\) and any function \(\nu\colon\mathscr{F}\to X\) satisfying $$\\nu(A\cup B)\nu(A)\nu(B)\\leq 1\quad\mbox{for disjoint }A,\, B\in\mathscr{F}$$ there is a vector measure \(\mu\colon\mathscr{F}\to X\) with \(\\nu(A)\mu(A)\\leq v\) for all \(A\in\mathscr{F}\). If this condition is valid when restricted to set algebras \(\mathscr{F}\) of cardinality less than some fixed cardinal number \(\kappa\), then we say that \(X\) has the \(\kappa\)\(\mathsf{SVM}\) property. The least cardinal \(\kappa\) for which \(X\) does not have the \(\kappa\)\(\mathsf{SVM}\) property (if it exists) is called the \(\mathsf{SVM}\) character of \(X\). We apply the machinery of twisted sums and quasilinear maps to characterise these properties and to determine \(\mathsf{SVM}\) characters for many classical Banach spaces. We also discuss connections between the \(\kappa\)\(\mathsf{SVM}\) property, \(\kappa\)injectivity and the 'threespace' problem.
 Almost orthogonally additive functions
Journal of Mathematical Analysis and Applications 400 (2013), 114.
Współautor: W. WyrobekKochanek
Abstrakt
If a function \(f\), acting on a Euclidean space \(\mathbb{R}^n\), is "almost" orthogonally additive in the sense that \(f(x+y)=f(x)+f(y)\) for all \((x,y)\in\bot\setminus Z\), where \(Z\) is a "negligible" subset of the \((2n1)\)dimensional manifold \(\bot\subset\mathbb{R}^{2n}\), then \(f\) coincides almost everywhere with some orthogonally additive mapping.
 Characterisation of \(L_p\)norms via Hölder's inequality
Journal of Mathematical Analysis and Applications 399 (2013), 403410.
Współautor: M. Lewicki
Abstrakt
We characterise \(L_p\)norms on the space of integrable step functions, defined on a probabilistic space, via Hölder's type inequality with an optimality condition.
 \(\mathcal{F}\)bases with brackets and with individual brackets in Banach spaces
Studia Mathematica 211 (2012), 259268.
Abstrakt
We provide a partial answer to the question of Vladimir Kadets whether given an \(\mathcal{F}\)basis of a Banach space \(X\), with respect to some filter \(\mathcal{F}\subset\mathcal{P}(\mathbb{N})\), the coordinate functionals are continuous. The answer is positive if the character of \(\mathcal{F}\) is less than \(\mathfrak{p}\). In this case every \(\mathcal{F}\)basis is an \(M\)basis with brackets which are determined by an element of \(\mathcal{F}\).
 On measurable solutions of a general functional equation on topological groups
Publicationes Mathematicae Debrecen 78 (2011), 527533.
Współautor: M. Lewicki
Abstrakt
We establish a theorem of the type "measurability implies continuity" for solutions \(f\) of the functional equation $$\Gamma(f(x),f(y))=\Phi(x,y,f(\alpha_1x+\beta_1y),\ldots ,f(\alpha_nx+\beta_ny))$$ under reasonable conditions upon the integers \(\alpha_i,\beta_i\) and the mappings \(\Gamma,\Phi\).

Probability distribution solutions of a general linear equation of infinite order II
Annales Polonici Mathematici 99 (2010), 215224.
Współautor: J. Morawiec
Abstrakt
Let \((\Omega,\mathcal{A},P)\) be a probability space and let \(\tau\colon\mathbb{R}\times\Omega\to\mathbb{R}\) be a mapping strictly increasing and continuous with respect to the first variable, and \(\mathcal{A}\)measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $$F(x)=\int_\Omega F(\tau(x,\omega))\, P(d\omega).$$We extend our uniquenesstype theorems obtained in Ann. Polon. Math. 95 (2009), 103114.

On a composite functional equation fulfilled by modulus of an additive function
Aequationes Mathematicae 80 (2010), 155172.
Abstrakt
We deal with the problem of determining general solutions \(f\colon\mathbb{R}\to\mathbb{R}\) of the following composite functional equation introduced by Fechner: $$f(f(x)f(y))=f(x+y)+f(xy)f(x)f(y).$$Our result gives a partial answer to this problem under some assumptions upon \(f(\mathbb{R})\). We are applying a theorem of Simon and Volkmann concerning a certain characterization of modulus of an additive function. A new proof of their result is also presented.

Stability aspects of arithmetic functions II
Acta Arithmetica 139 (2009), 131146.
Abstrakt
We deal with the stability problem for arithmetic additive and multiplicative functions in the sense of Hyers and Ulam. In the additive case, the main stability result has the following form: Let an arithmetic function \(f\colon\mathbb{N}\to\mathbb{R}\) satisfy the condition $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)f(x)f(y)\vert\leq\varepsilon.$$Then there exists a real additive function \(\widetilde{f}\) such that \(\vert f(x)\widetilde{f}(x)\vert\leq K\varepsilon\) for all \(x\in\mathbb{N}\) with the KaltonRoberts constant \(K\leq 89/2\).
In the multiplicative case, the main stability assertion is similar: Let an arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) satisfy the condition $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\Biggl\frac{f(xy)}{f(x)f(y)}1\Biggr\leq\varepsilon$$with some \(0\leq\varepsilon<1\). Then there exists an arithmetic multiplicative function \(\widetilde{f}\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) such that $$\Biggl\frac{f(x)}{\widetilde{f}(x)}1\Biggr\leq\delta(\varepsilon)\quad\mbox{and}\quad\Biggl\frac{\widetilde{f}(x)}{f(x)}1\Biggr\leq\delta(\varepsilon)\quad\mbox{for }x\in\mathbb{N},$$
where \(\delta(\varepsilon)\) is a nonnegative number depending only on \(\varepsilon\). Moreover, \(\delta(\varepsilon)\to 0\) as \(\varepsilon\to 0\).
Unfortunately, as it was pointed out by Professor Andrzej Schinzel, there is a gap in the proof of the first mentioned result, which also impacts the second. In the subsequent work we show that these statements hold true under some additional requirements. Therefore, the two theorems mentioned above have a hypothetical nature at the moment.
Corrigendum to "Stability aspects of arithmetic functions II" (Acta Arith. 139 (2009), 131146)
Acta Arithmetica 149 (2011), 8398.
Abstrakt
The main results from the above paper are corrected. For instance, in the additive case the following theorem is proved: Let \(f\colon\mathbb{N}\to\mathbb{R}\) be an arithmetic function satisfying $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)f(x)f(y)\vert\leq\varepsilon$$ and $$x,y\in\mathbb{N},\,\{p\in\mathbb{P}\colon p\mid x\}=\{p\in\mathbb{P}\colon p\mid y\}\,\,\Rightarrow\,\,\vert f(x)f(y)\vert\leq 2\varepsilon.$$ Then there exists a real strongly additive function \(\widetilde{f}\) such that \(\vert f(x)\widetilde{f}(x)\vert\leq K^\ast\varepsilon\) for all \(x\in\mathbb{N}\) with some absolute constant \(K^\ast\leq 89/2\).
 Probability distribution solutions of a general linear equation of infinite order
Annales Polonici Mathematici 95 (2009), 103114.
Współautor: J. Morawiec
Abstrakt
Let \((\Omega,\mathcal{A},P)\) be a probability space and let \(\tau\colon\mathbb{R}\times\Omega\to\mathbb{R}\) be strictly increasing and continuous with respect to the first variable, and \(\mathcal{A}\)measurable with respect to the second variable. We obtain a partial characterisation and a uniquenesstype result for solutions of the general linear equation $$F(x)=\int_\Omega F(\tau(x,\omega))\, P(d\omega)$$ in the class of probability distribution functions.

An inconsistency equation involving means
Colloquium Mathematicum 115 (2009), 8799.
Współautor: R. Ger
Abstrakt
We show that any quasiarithmetic mean \(A_\varphi\) and any nonquasiarithmetic mean \(M\) (reasonably regular) are inconsistent in the sense that the only solutions \(f\) of both equations $$f(M(x,y))=A_\varphi(f(x),f(y))$$ and $$f(A_\varphi(x,y))=M(f(x),f(y))$$ are the constant ones.

Measurable orthogonally additive functions modulo a discrete subgroup
Acta Mathematica Hungarica 123 (2009), 239248.
Współautor: W. Wyrobek
Abstrakt
Under appropriate conditions on Abelian topological groups \(G\) and \(H\), an orthogonality \(\bot\subset G^2\) and a \(\sigma\)algebra \(\mathfrak{M}\) of subsets of \(G\) we decompose an \(\mathfrak{M}\)measurable function \(f\colon G\to H\) which is orthogonally additive modulo a discrete subgroup \(K\) of \(H\) into its continuous additive and continuous quadratic part (modulo \(K\)).

Stability aspects of arithmetic functions
Acta Arithmetica 132 (2008), 8798.
Abstrakt
An arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\) is called almost additive if for some fixed \(\varepsilon\geq 0\) we have:
$$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)f(x)f(y)\vert\leq\varepsilon.$$
An arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) is called almost multiplicative if for some \(\varepsilon\in [0,1)\) we have: $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\Biggl\frac{f(xy)}{f(x)f(y)}1\Biggr\leq\varepsilon.$$Using the Banach limit technique and Ramsey's theorem we derive some stability properties for almost additive and almost multiplicative functions. For instance, we show that for any almost additive function \(f\colon\mathbb{N}\to\mathbb{R}\) satisfying $$\liminf_{x\to\infty}(f(x+1)f(x))\geq 0$$there exists a constant \(c\in\mathbb{R}\) such that $$\vert f(x)c\log x\vert\leq\varepsilon,\quad x\in\mathbb{N}.$$This assertion generalizes some earlier results due to P. Erdös, I. Kátai and A. Máté.

Stability problem for numbertheoretically multiplicative functions
Proceedings of the American Mathematical Society 135 (2007), 25912597.
Współautor: M. Lewicki
Abstrakt
We deal with the stability question for multiplicative mappings in the sense of number theory. It turns out that the conditional stability assumption: $$\vert f(xy)f(x)f(y)\vert\leq\varepsilon\quad\mbox{for relatively prime }x,y$$ implies that \(f\) lies near to some numbertheoretically multiplicative function. The domain of \(f\) can be general enough to admit, in special cases, the reduction of our result to the well known J.A. BakerJ. LawrenceF. Zorzitto superstability theorem.