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Niels Jakob Laustsen

Publications and source records attributed to Niels Jakob Laustsen.

At least 19 recordsLinked to original sources
arXiv2026

Maximal right ideals of the Banach algebra of bounded operators on a Banach space

We study finitely generated maximal right ideals of the Banach algebra $\mathcal{B}(E)$ of bounded operators on a complex Banach space $E$. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when $E$ is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals $\operatorname{Lift}(T)=\{TU:U\in\mathcal{B}(E,E^n)\}$, where $T\in\mathcal{B}(E^n,E)$ for some $n\in\mathbb{N}$, we identify the exact operator-theoretic obstruction. The ideal $\operatorname{Lift}(T)$ contains the finite-rank operators precisely when $T$ is surjective, and it equals $\mathcal{B}(E)$ precisely when $T$ is right invertible. If $T$ is surjective but not right invertible, then $\operatorname{Lift}(T)$ is maximal exactly when the row operator $[T\ S]$ is right invertible for every $S\in\mathcal{B}(E)\setminus\operatorname{Lift}(T)$. We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of $\ell_1$, KB-spaces, Lebesgue spaces $L_p(μ)$ for $1\leqslant p<\infty$, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, $\ell_1(Γ)$-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces $\ell_1(Γ)\oplus H$ with $H$ a separable Hilbert space.

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arXiv2025

Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces

Let $X$ be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces $B_p$ for $1<p<\infty$; (ii) the $p$-convexified Schreier spaces $S_p$ for $1\le p<\infty$; (iii) the sequence spaces $\ell_p$ for $1\le p<\infty$ (and $c_0$). We show that the quotient algebra of strictly singular by compact operators on $X$ is nilpotent; that is, there is a natural number $k$, dependent only on the collections of direct summands from each of the three families, such that: - every composition of $k+1$ strictly singular operators on $X$ is compact; - there are $k$ strictly singular operators on $X$ whose composition is not compact.

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arXiv2025

Strictly singular operators on the Baernstein and Schreier spaces

Every composition of two strictly singular operators is compact on the Baernstein space $B_p$ for $1 < p < \infty$ and on the $p$-convexified Schreier space $S_{p}$ for $1 \leq p < \infty$. Furthermore, every subsymmetric basic sequence in $B_p$ (respectively, $S_p$) is equivalent to the unit vector basis for $\ell_p$ (respectively, $c_0$), and the Banach spaces $B_p$ and $S_p$ contain block basic sequences whose closed span is not complemented.

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arXiv2025

Kernels of Bounded Operators on the Classical Transfinite Banach Sequence Spaces

Every closed subspace of each of the Banach spaces $X = \ell_p(Γ)$ and $X=c_0(Γ)$, where $Γ$ is a set and $1<p<\infty$, is the kernel of a bounded operator $X\to X$. On the other hand, whenever $Γ$ is an uncountable set, $\ell_1(Γ)$ contains a closed subspace that is not the kernel of any bounded operator $\ell_1(Γ)\to\ell_1(Γ)$.

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arXiv2024

Closed ideals of operators on the Baernstein and Schreier spaces

We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces $B_p$ for $1<p<\infty$ and the Schreier spaces $S_p$ for $1\le p<\infty$. Our main conclusion is that there are $2^{\mathfrak{c}}$ many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also $2^{\mathfrak{c}}$ many closed ideals that contain projections of infinite rank. Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the higher-order Schreier spaces play a key role in the proofs, as does the Johnson-Schechtman technique for constructing $2^{\mathfrak{c}}$ many closed ideals of operators on a Banach space.

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arXiv2024

Kernels of operators on Banach spaces induced by almost disjoint families

Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$Γ$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(Γ)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$. We show that if~$\mathcal{A}$ has cardinality greater than~$Γ$, then the closed subspace of~$X_{\mathcal{A}}$ spanned by the indicator functions of sets of the form $\bigcap_{j=1}^{n+1}A_j$, where $n\in\N$ and $A_1,\ldots,A_{n+1}\in\mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\rightarrow \ell_{\infty}(Γ)$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{\infty}(Γ) : \text{the set}\ \{γ\in Γ: \lvert x(γ)\rvert > \varepsilon \}\ \text{has cardinality smaller than}\ Γ \text{for every}\ \varepsilon>0\bigr\} \] of~$\ell_\infty(Γ)$ is not the kernel of any bounded operator on~$\ell_\infty(Γ)$; this generalises results of Kalton and of Pełczyński and Sudakov. The situation is more complex for the Banach space~$\ell_\infty^c(Γ)$ of countably supported, bounded functions defined on an uncountable set~$Γ$. We show that it is undecidable in \textsf{ZFC} whether every bounded operator on~$\ell_\infty^c(ω_1)$ which vanishes on~$c_0(ω_1)$ must vanish on a subspace of the form~$\ell_\infty^c(A)$ for some uncountable subset~$A$ of~$ω_1$.

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arXiv2024

Banach lattices of positively homogeneous functions induced by a Banach space

Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space $E$. The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space $E$ is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

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arXiv2023

Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

We show that for each of the following Banach spaces~$X$, the quotient algebra $\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed ideal $\mathscr{I}$ of $\mathscr{B}(X)\colon$ - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}$\quad and its dual,\quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}$, - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}\oplus c_0(Γ)$\quad and its dual, \quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}\oplus\ell_1(Γ)$,\quad for an uncountable cardinal number~$Γ$, - $X = C_0(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing at infinity on the locally compact Mrówka space~$K_{\mathcal{A}}$ induced by an uncountable, almost disjoint family~$\mathcal{A}$ of infinite subsets of~$\mathbb{N}$, constructed such that $C_0(K_{\mathcal{A}})$ admits "few operators". Equivalently, this result states that every homomorphism from~$\mathscr{B}(X)$ into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in $\mathscr{B}(X)\setminus\mathscr{I}$ with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.

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arXiv2023

Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces

Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathcal{Y})$, either no pair of Fredholm operators of index~$k$ acting on $\mathcal{X}$ and $\mathcal{Y}$, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, the question whether EAE and SC coincide for Fredholm operators of index~$k$ depends only on the geometry of the underlying Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, not on the properties of the operators themselves. We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC, and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al, which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.

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arXiv2022

Approximately multiplicative maps between algebras of bounded operators on Banach spaces

We show that for any separable reflexive Banach space $X$ and a large class of Banach spaces $E$, including those with a subsymmetric shrinking basis but also all spaces $L_p$ for $1\leq p \leq \infty$, every bounded linear map ${\mathcal B}(E)\to {\mathcal B}(X)$ which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism ${\mathcal B}(E)\to {\mathcal B}(X)$. That is, the pair $({\mathcal B}(E), {\mathcal B}(X))$ has the AMNM property in the sense of Johnson (\textit{J.~London Math.\ Soc.} 1988). Previously this was only known for $E=X=\ell_p$ with $1<p<\infty$; even for those cases, we improve on the previous methods and obtain better constants in various estimates. A crucial role in our approach is played by a new result, motivated by cohomological techniques, which establishes AMNM properties relative to an amenable subalgebra; this generalizes a theorem of Johnson (\textit{op cit.}).

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arXiv2021

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $Ψ$-space and Isbell--Mrówka space. We construct an uncountable, almost disjoint family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded linear operator on $C_0(K_{\mathcal A})$ is the sum of a scalar multiple of the identity and an operator that factors through $c_0$ (which in this case is equivalent to having separable range). This implies that $C_0(K_{\mathcal A})$ has the fewest possible decompositions: whenever $C_0(K_{\mathcal A})=X\oplus Y$ with $dim({X})=\infty$, $dim({Y})=\infty$, either ${X}$ is isomorphic to $C_0(K_{\mathcal A})$ and ${Y}$ to $c_0$, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space $C_0(K_{\mathcal A})$ concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms. To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family of Mrówka we need to deal with $\mathbb N \times \mathbb N$-matrices rather than with the usual partitioners. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.

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arXiv2021

Free Banach lattices under convexity conditions

We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.

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arXiv2020

Closed ideals of operators on the Tsirelson and Schreier spaces

Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empty subset $N$ of $\mathbb N$. A closed ideal of $\mathscr{B}(X)$ is spatial if it is generated by $P_N$ for some $N$. We can now state our main conclusions as follows: i) the family of spatial ideals lying strictly between the ideal of compact operators and $\mathscr{B}(X)$ is non-empty and has no minimal or maximal elements; ii) for each pair $\mathscr{I}\subsetneqq\mathscr{J}$ of spatial ideals, there is a family $\{Γ_L\colon L\in Δ\}$, where the index set $Δ$ has the cardinality of the continuum, such that $Γ_L$ is an uncountable chain of spatial ideals, $\bigcupΓ_L$ is a closed ideal that is not spatial, and $$ \mathscr{I}\subsetneqq\mathscr{L}\subsetneqq\mathscr{J}\qquad\text{and}\qquad \overline{\mathscr{L}+\mathscr{M}} = \mathscr{J}$$ whenever $L,M\inΔ$ are distinct and $\mathscr{L}\inΓ_L$, $\mathscr{M}\inΓ_M$.

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arXiv2019

Vector lattices admitting a positively homogeneous continuous function calculus

We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in\mathbb N$: - there is a vector lattice homomorphism $Φ_{\boldsymbol{x}}\colon H_n\to X$ such that $Φ_{\boldsymbol{x}}(π_i^{(n)})=x_i$ $(i\in\{1,\ldots,n\})$, where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on $\mathbb R^n$ and $π_i^{(n)}\colon\mathbb R^n\to\mathbb R$ is the $i^{\text{th}}$ coordinate projection; - there is a positive element $e\in X$ such that $e\geqslant\lvert x_1\rvert\vee\cdots\vee\lvert x_n\rvert$ and the norm $\lVert x\rVert_e = \inf\bigl\{λ\in[0,\infty)\:\colon\:\lvert x\rvert\leλe\bigr\}$, defined for each $x$ in the order ideal $I_e$ of $X$ generated by $e$, is complete when restricted to the closed sublattice of $I_e$ generated by $x_1,\ldots,x_n$. Moreover, we show that a vector space which admits a `sufficiently strong' $H_n$-function calculus for each $n\in\mathbb N$ is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function calculus, while others do not.

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arXiv2018

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space $E$ which contains a closed subspace that cannot be realized as the kernel of any bounded operator on $E$. This implies that the Banach algebra $\mathcal{B}(E)$ of bounded operators on $E$ fails to be weak*-topologically left Noetherian. The Banach space $E$ that we use is the dual of Wark's non-separable, reflexive Banach space with few operators.

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arXiv2017

Factorization of the identity through operators with large diagonal

Given a Banach space~$X$ with an unconditional basis, we consider the following question: does the identity on~$X$ factor through every operator on~$X$ with large diagonal relative to the unconditional basis? We show that on Gowers' unconditional Banach space, there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces $H^p(H^q)$, where $1 \leq p,q < \infty$, with the bi-parameter Haar system, this problem always has a positive solution. The spaces $L^p, 1 < p < \infty$, were treated first by Andrew~[{\em Studia Math.}~1979].

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