An arXiv preprint lays out a way to use numerical ranges - sets of complex values built from an operator or relation - to fence in spectra of multivalued operators. Under the paper's stated assumptions, it also gives a distance-based bound on the size of a resolvent, the inverse object used away from the spectrum. The work adapts the idea to linear relations in Banach spaces, where one input may be linked to more than one output.
The adjoint matters
The framework separates three kinds of spectral information. For a closed linear relation, the point spectrum is contained in its numerical range, the approximate point spectrum is contained in the range's closure, and the full spectrum is contained in that closure together with the numerical range of the Banach adjoint. In ordinary terms, the test covers exact eigenvalue cases, near-eigenvalue cases and the full spectrum, with the adjoint included where the broadest enclosure is needed.
One constructed example makes that caveat concrete: the relation's numerical range is the single value zero, while its spectrum is the entire complex plane. Combining the relation's range with the adjoint's range gives the entire complex plane, restoring the enclosure.
When the relation has full domain, the result simplifies: for a closed relation, its spectrum is contained in the closure of its own numerical range. The paper also gives a resolvent estimate outside that closed range: for a point in the resolvent, the resolvent norm is no more than one divided by the distance to the numerical range.
A tighter test for operator pencils
The same machinery is carried over to operator pencils, parameter-dependent operator expressions. For the relation associated with a pencil, the pencil's point spectrum lies in the relation's numerical range, while its full spectrum lies in the closure of that range together with the adjoint range.
Two Banach-space numerical ranges defined for a pencil are nested: the range denoted w(A,E) is contained in the range denoted W(A,E). A separate result encloses the pencil spectrum using the numerical ranges of the pencil and its Banach adjoint.
In a finite-dimensional example, the relation-based enclosure is exact: the spectrum, the relation's numerical range and the adjoint's numerical range all collapse to a single value, denoted a. The classical pencil ranges instead form an unbounded ray starting at a and extending in the nonnegative real direction. That sharper result belongs to the constructed example; it does not establish the same behavior for every operator pencil.
The examples diverge over time
The final illustration turns to semigroups, mathematical families used here to track evolution over time. It compares p=1 and p=2 on the interval (0,1). In the p=1 case, the numerical abscissa, a range-based growth indicator, is minus one for B1, and the generated semigroup has an exact norm factor of e to the minus t for every nonnegative time.
The p=2 example starts differently. Its norm initially grows, and the displayed calculation gives no decay estimate with any nonnegative alpha, even though the example is exponentially stable. Within these constructions, the contrast illustrates how a Banach-space numerical abscissa can provide decay information that the corresponding Hilbert-space numerical range does not.
What the examples do not settle
These are conditional theorems and constructed demonstrations, not a universal verdict. The operator-pencil results rely on the paper's assumptions about boundedness, closedness, density and a nonempty resolvent, while the finite-dimensional and semigroup examples do not establish that the same sharper enclosures or decay conclusions hold for every pencil or semigroup.
The document is an arXiv preprint, version 1, dated 20 August 2026.
Paper data and sources
Original title: On the Numerical Range of Linear Relations in Banach Spaces
Authors: Wissal Boubaker, Hannes Gernandt, Wafa Selmi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text