Norms of multiplication operators: answering Fialkow--Loebl question

Abstract: Let $\mathcal{M}$ be a factor equipped with a semi-finite faithful normal trace $τ$. Let $E(0,\infty)$ be a symmetrically normed function space and $E(\mathcal{M},τ)$ be the corresponding symmetrically normed operator space. Suppose that $a, b$ are $τ$-measurable operators affiliated with $\mathcal{M}$. It is shown that the range of the multiplication operator $S_{a,b}: x\mapsto axb$ on $\mathcal{M}$ is contained in $E(\mathcal{M}, τ)$ if and only if $μ(a)μ(b)$ belongs to $E(0, \infty)$, where $μ(x)$ stands for the generalized singular value function of a $τ$-measurable operators $x$ affiliated with $\mathcal{M}$. Moreover, we have $$|S_{a,b}|<em>{\mathcal{M}\to E(\mathcal{M},τ)}= | μ(a )μ( b) |</em>{E (0,\infty) },$$ which answers a question by Fialkow and Loebl (1984). We also consider the quasi-normed case, and show that the natural quasi-norm of weak $L_p$-space, $0&lt;p&lt;\infty$, is not monotone with respect to the logarithmic submajorisation.

• The paper proves that the range and norm of S_{a,b}(x)=axb are characterized exactly by the product of generalized singular values, eliminating the earlier factor-of-two bound for every symmetrically normed ideal over a semi-finite factor.
• The authors combine uniform submajorisation, logarithmic submajorisation, spectral decompositions, and partial-isometry constructions to establish matching upper and lower bounds, extending the result beyond compact operators and Schatten classes.
• The quasi-normed extension applies under log-monotonicity, while weak-L_p quasi-norms provide counterexamples showing that the exact formula can fail without this condition and identifying key directions for future research.

Overview and main result

This paper by Huang, Sukochev, Xu, and Zhu resolves a question posed by Fialkow and Loebl in 1984 concerning the norm of multiplication operators on ideals of compact operators. Let $\mathcal{M}$ be a factor with semi-finite faithful normal trace $\tau$, let $E(0,\infty)$ be a symmetrically normed function space, and let $E(\mathcal{M},\tau)$ be the associated symmetrically normed operator space of $\tau$-measurable operators. For $a,b \in S(\mathcal{M},\tau)$, define the multiplication operator $S_{a,b}(x) = axb$ on $\mathcal{M}$. The main theorem establishes that

$$\mathrm{Ran}(S_{a,b}) \subseteq E(\mathcal{M},\tau) \iff \mu(a)\mu(b) \in E(0,\infty),$$

and, moreover,

$$\|S_{a,b}\|_{\to E(\mathcal{M},\tau)} = \|\mu(a)\mu(b)\|_{E(0,\infty)},$$

where $\tau$0 denotes the generalized singular value function. In the special case $\tau$1, this answers the Fialkow–Loebl question affirmatively for all symmetric norm ideals, removing both the compactness assumptions on $\tau$2 and $\tau$3 and the structural restrictions on the ideal imposed in the classical treatments of Schatten, Gohberg–Krein, and Simon.

The original 1984 result gave only the two-sided estimate

$\tau$4

with equality known only for Schatten classes $\tau$5, $\tau$6. The present paper eliminates the factor of 2 and extends the identity to arbitrary symmetric norms.

Method of proof

The proof combines several ingredients. The upper bound rests on Sukochev's uniform submajorisation inequality $\tau$7 together with monotonicity of symmetric norms under uniform submajorisation; this replaces the Schatten-specific inequality used by Fialkow–Loebl, which does not generalize beyond $\tau$8-type spaces.

The lower bound is the harder direction and occupies most of the technical work. Two key lemmas construct, for arbitrary $\tau$9, partial isometries $E(0,\infty)$0 satisfying

$E(0,\infty)$1

The construction splits according to whether the operators have vanishing essential singular values ($E(0,\infty)$2 or $E(0,\infty)$3), handled via an approximation argument based on decomposing $E(0,\infty)$4 into layers where it varies slowly (within a factor $E(0,\infty)$5), and the remaining case ($E(0,\infty)$6, $E(0,\infty)$7), handled by splitting into a "compact-like" part and an infinite-dimensional part where the operator is bounded below on an infinite projection. The factor hypothesis is essential here: total comparability of projections (Kadison–Ringrose) supplies the partial isometries intertwining spectral projections of $E(0,\infty)$8 and $E(0,\infty)$9. The polar decomposition reduces the general case to positive operators, since $E(\mathcal{M},\tau)$0.

The quasi-normed case

The authors then extend the framework to symmetrically quasi-normed spaces. The analogue of the main theorem holds whenever the quasi-norm is monotone with respect to logarithmic submajorisation $E(\mathcal{M},\tau)$1, using the Weyl-type inequality $E(\mathcal{M},\tau)$2 valid on $E(\mathcal{M},\tau)$3. This covers Lorentz spaces $E(\mathcal{M},\tau)$4 for all $E(\mathcal{M},\tau)$5, weighted Lorentz spaces, and noncommutative $E(\mathcal{M},\tau)$6-spaces for all $E(\mathcal{M},\tau)$7, thereby extending the Fialkow–Loebl Schatten-class formula to the full quasi-normed range.

Two auxiliary results of independent interest are established:

• Geometric stability: every quasi-Banach symmetric function space is geometrically stable, unifying earlier results of Kalton (quasi-Banach ideals) and Fack (Banach symmetric spaces).
• Equivalent log-monotone quasi-norm: every symmetrically quasi-normed function space admits an equivalent quasi-norm that is monotone with respect to $E(\mathcal{M},\tau)$8, extending a proposition of Fack. Consequently, for every symmetrically quasi-normed $E(\mathcal{M},\tau)$9 there is a constant $\tau$0 depending only on $\tau$1 such that

$\tau$2

A negative example: weak $\tau$3 quasi-norms

A notable counterpoint shows the exact formula fails without log-monotonicity. Using the optimal constants in the Hölder inequality for weak-$\tau$4 quasi-norms obtained by Sukochev–Zanin, the authors exhibit $\tau$5 with $\tau$6 such that

$\tau$7

since the sharp constant $\tau$8 exceeds 1 while $\tau$9. Hence the natural quasi-norm of $a,b \in S(\mathcal{M},\tau)$0 is not monotone with respect to logarithmic submajorisation — a fact the authors describe as unexpected, given that $a,b \in S(\mathcal{M},\tau)$1 possesses an equivalent fully symmetric norm for $a,b \in S(\mathcal{M},\tau)$2. This also implies the converse of the range-inclusion lemma fails in general quasi-normed settings (trivially so when $a,b \in S(\mathcal{M},\tau)$3 with $a,b \in S(\mathcal{M},\tau)$4).

Limitations and open questions

Several points remain open. The main theorem requires $a,b \in S(\mathcal{M},\tau)$5 to be a factor; extension to general semi-finite von Neumann algebras, where projections are not totally comparable, is not addressed. The counterexample leaves open whether the Hölder-type inequalities of Dodds–Dodds–Sukochev–Zanin hold without the log-monotonicity assumption, and how to characterize linear isometries on commutative and noncommutative weak-$a,b \in S(\mathcal{M},\tau)$6 spaces equipped with their natural quasi-norms. Finally, it remains unknown whether there exists a symmetric function space whose norm is monotone with respect to $a,b \in S(\mathcal{M},\tau)$7 but not fully symmetric; such examples exist for sequence spaces, but the function-space case posed by Dodds et al. is unresolved.

Conclusion

The paper settles the 1984 Fialkow–Loebl problem by proving the exact identity $a,b \in S(\mathcal{M},\tau)$8 for multiplication operators acting into any symmetrically normed operator space over a semi-finite factor, and its quasi-normed analogue under log-monotonicity. The accompanying negative result for weak-$a,b \in S(\mathcal{M},\tau)$9 quasi-norms delineates precisely where the identity breaks down, identifying log-monotonicity as the operative hypothesis rather than completeness or full symmetry.