What the study found
The study shows that for a complex self-adjoint matrix, the Schur multiplier norm can be determined by a minimization formula involving a diagonal bound. It also gives a corresponding formula for the dual norm of the Schur multiplier norm.
Why the authors say this matters
The authors say they study the dual norm minimization problem as a formal linear program, which suggests a way to frame the problem in optimization terms. No broader application is stated in the abstract.
What the researchers tested
The paper works with complex self-adjoint n × n matrices and defines a dual norm for norms on M(m,n)(C), the space of complex m × n matrices. It introduces diagonal matrices Δ(λ) built from vectors λ in C^n and uses inequalities involving these matrices to characterize the norms.
What worked and what didn't
For a self-adjoint matrix A, the Schur multiplier norm is given by the minimum of the infinity norm of diag(P) over all P such that -P ≤ A ≤ P. For the dual norm, the paper gives the minimum of Tr_n(Δ(λ)) over real vectors λ satisfying -Δ(λ) ≤ A ≤ Δ(λ). The abstract does not report any failed cases or comparisons.
What to keep in mind
The available summary does not describe limitations, examples, or numerical experiments. It also does not state any application beyond the formal linear-program formulation of the minimization problem.
Key points
- The Schur multiplier norm of a complex self-adjoint matrix is expressed by a minimization formula.
- The dual norm of the Schur multiplier norm is also given by a minimization formula.
- The dual norm formula uses real vectors λ and diagonal matrices Δ(λ).
- The paper studies the minimization problem as a formal linear program.
- The abstract does not describe limitations, examples, or applications beyond the optimization framing.
Disclosure
- Research title:
- Schur multiplier norm and its dual are expressed by minimization formulas
- Authors:
- Erik Christensen
- Institutions:
- University of Copenhagen
- Publication date:
- 2026-04-23
- OpenAlex record:
- View
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