Abstract We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $$L^p$$ L p -potentials on bounded and unbounded Lipschitz domains with compact boundaries.

The main result of this paper are novel two-sided estimates of the essential resolvent norm for closed linear operators T . We prove that the growth of \|(T-\lambda)^{-1}\|_{\textup{e}} is governed by the distance of a point \lambda\in{}\rho(T){}\setminus{}W_{\textup{e}}(T) to the essential numerical range W_{\textup{e}}(T) . We extend these bounds even to points \lambda{}\in \mathbb{C}\setminus W_{\textup{e}}(T) outside the resolvent set \rho(T) with (T{}-{}\lambda)^{-1} replaced by the Moore–Penrose resolvent (T-\lambda)^{\dagger}{} . We use similar ideas to prove essential growth bounds in terms of the real part of the essential numerical range of generators of C_{0} -semigroups. Further, we study the essential approximate point spectrum \sigma_{{\textup{eap}}}(T) and the essential minimum modulus \gamma_{\textup{e}}(T) , in particular, their relations to the various essential spectra and the essential norm of the Moore–Penrose inverse, respectively. An important consequence of our results are new perturbation results for the spectra and essential spectra (of type 2) for accretive and sectorial T . Applications e.g. to Schrödinger operators with purely imaginary rapidly oscillating potentials in \mathbb{R}^{d} illustrate our results.

ABSTRACTAimGenetic diversity is key to the long‐term maintenance and adaptability of species to changing environments. While for above‐ground ecosystems, the monitoring and understanding of genetic diversity has advanced substantially, some less accessible ecosystems and their organisms have been largely overseen. This is particularly the case for groundwater organisms. It is not only difficult to collect sufficient genetic data to identify spatial patterns but they may also have been experiencing very different drivers to population size, occurrence, and genetic structure due to very limited dispersal capacity and persistence in areas extending glacial cycles.LocationSwitzerland.MethodsHere, we use a unique and spatially highly resolved data set containing a representative collection of thousands of Niphargus amphipod individuals across Switzerland. We analysed the genetic structure of ~1300 individuals of five species within their contemporary distribution.ResultsWe found a significantly higher diversity in karstic aquifers and a correlation between the genetic diversity of a species and the proportion of its distribution in the karst.Main ConclusionsThis identifies karstic ecosystems as specific targets for future conservation programmes but also indicates that these karstic areas could have been possible refugia of Pleistocene persistence. Being epicentres of genetic diversity, the protection of the karst is also central for the maintenance of groundwater organisms' adaptive potential to future climatic changes.

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem.

In this paper we consider a class of unbounded Toeplitz operators with rational matrix symbols that have poles on the unit circle and employ state space realization techniques from linear systems theory, as used in our earlier analysis in Groenewald et al. (J Math Anal Appl 532, Paper no. 127925, 2024) of this class of operators, to study the connection with semi-infinite Toeplitz matrices and to determine the essential spectrum and resolvent set.

Assume that A, A1, and A2 are three selfadjoint linear relations (multi-valued linear operators) in a certain complex Hilbert space. In this study, conditions are presented for the multi-valued operator equality A=A1A2 to hold when the inclusion A⊂A1A2 is assumed to be satisfied. The present study is strongly motivated by the invalidity of a classical result from A. Devinatz, A. E. Nussbaum, and J. von Neumann in the general case of selfadjoint linear relations. Two types of conditions for the aforementioned equality to hold are presented. Firstly, a condition is given in terms of the resolvent sets of the involved objects, which does not depend on the product structure of the right-hand side, A1A2. Secondly, a condition is also presented where the structure of the right-hand side is taken into account. This one is based on the notion of the L-stability of a linear operator under linear subspaces. It should be mentioned that the classical Devinatz–Nussbaum–von Neumann theorem is obtained as a particular case of one of the main results.

We introduce the concept of a convex bounded operator and we define the convex norm for convex bounded linear operator. The aim of this research is to introduce the properties of the spectrum and resolvent sets of convex bounded linear operators on convex normed spaces.

Conditions for a linear closed operator are obtained in terms of the location of its resolvent set and estimates for its resolvent and its derivatives, which are necessary and sufficient to generate a strongly continuous resolving family of operators by this operator. Some properties of such resolving families are proved, and a theorem on the unique solvability of the Cauchy problem for the corresponding linear inhomogeneous equation is obtained. The results are used to prove the unique solvability of initial boundary value problems for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables and with a distributed derivative in time.

Connected graphs G. W = {w1, w2,..., wk} is a subset of V with a predetermined order. The line vector r(v|W) = (d(v, w1), d(v, w2),..., d(v, wk)) is the measurement depicting v with regards to W for each v ∈ V. If V's vertex utilize various metrics, W resolves G. Their fundamental magnitude, dim(G), is their lowest cardinality. A resolved set W is non-isolated if its influenced subsection ⟨W ⟩ has no single vertex. The simplest connection of a non-isolated resolved set of G is nr. An nr-set for G is a non-isolated resolution set of cardinality nr(G). In this study, we prove that the chart G has a unique nr-set. We also build a 2n-vertex graph G using nr-set. W which means nr(G) = n and r(vi|W) = (1,...2, 1), where 2 is in the ith location, represents every vertex not in W. Further we established the nr-value for the highly irregular graph Hn,n and for the Wheel Wn. Also we determined the nr-value for corona product of some graphs.

Abstract The across‐shore transport of meroplanktonic larvae is predominantly driven by coastal physical processes, resulting in episodic recruitment of benthic species. Historically, due to the sampling challenges associated with resolving these advective mechanisms across the continental shelf, relevant components of larval transport have been difficult to isolate and understand. We use three‐dimensional temperature and velocity data from an array of 29 moorings to identify fundamental physical processes that could have generated successful across‐shore transport and settlement of meroplankton. The dense spatial and temporal sampling from this array allows us to use Lagrangian particle tracking to estimate the influences of wind conditions and the internal tide on the across‐shore transport of planktonic larvae. Settlement was found to be episodic at all depths studied. Above mid‐water, modeled larvae were successfully transported onshore by the internal tide during wind relaxations. Surprisingly, abundant pulses of shallow‐water larvae were supplied to the coast on occasions when strong, upwelling‐favorable winds (> 4 m s−1) drove offshore‐flowing surface waters, revealing a complex, potentially topographically influenced flow. These intense upwelling‐favorable winds also contributed to subsurface onshore flows that created large pulses of larval settlement in deeper waters (> 20 m). Our analyses from this highly resolved data set provide novel insights into the interactions of physical drivers in creating episodic pulses of coastal larval recruitment.

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

An equivalent norm in the weighted Bergman space A ω p , induced by an ω in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood–Paley inequalities are also considered. On the way to the proofs, we characterize the q-Carleson measures for the weighted Bergman space A ω p and the boundedness of a Hörmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators T g (f)(z)=∫ 0 z g ′ (ζ)f(ζ)dζ acting on A ω p .

Abstract AimAs climate change increases the frequency and severity of droughts in many regions, conservation during drought is becoming a major challenge for ecologists. Droughts are multidimensional climate events whose impacts may be moderated by changes in temperature, water availability or food availability, or some combination of these. Simultaneously, other stressors such as extensive anthropogenic landscape modification may synergize with drought. Useful observational models for guiding conservation decision‐making during drought require multidimensional, dynamic representations to disentangle possible drought impacts, and consequently, they will require large, highly resolved data sets. In this paper, we develop a two‐stage predictive framework for assessing how drought impacts vary with species, habitats and climate pathways.LocationCentral Valley, California, USA.MethodsWe used a two‐stage counterfactual analysis combining predictive linear mixed models and N‐mixture models to characterize the multidimensional impacts of drought on 66 bird species. We analysed counts from the eBird participatory science data set between 2010 and 2019 and produced species‐ and habitat‐specific estimates of the impact of drought on relative abundance.ResultsWe found that while fewer than a quarter (16/66) of species experienced abundance declines during drought, nearly half of all species (27/66) changed their habitat associations during drought. Among species that shifted their habitat associations, the use of natural habitats declined during drought while use of developed habitat and perennial agricultural habitat increased.Main ConclusionsOur findings suggest that birds take advantage of agricultural and developed land with artificial irrigation and heat‐buffering microhabitat structure, such as in orchards or parks, to buffer drought impacts. A working lands approach that promotes biodiversity and mitigates stressors across a human‐induced water gradient will be critical for conserving birds during drought.

Results of an omega-order preserving partial contraction mapping (omega-OCPn) in generalized spaces are presented in this study. Assumed to be a closed linear operator on a Banach space X with a non-empty resolvent set rho(A) is A in omega-OCPn. If A is densely defined, the extrapolation spaces X-1 and X-1 will be associated with A in agreement. However, X-1 is a proper closed subspace of X-1 if A is not densely defined. Then, we demonstrated that the reason these spaces exist is because (X*)-1 and D(A0) are naturally isomorphic to (X*)-1 and (X*)-1, respectively.

Abstract. Let be a strongly continuous, one-parameter semigroup of contractions on a complex Hilbert space 𝐻. The generator of is the linear operator with domain defined by Let be a closed densely defined operator on a Hilbert space with domain . For we define to be the set of all for which there exists a neighborhood of with analytic on having values in such that for all . This set is open and contains the resolvent set of . By definition, the local spectrum of at denoted by is the complement of , so it is a closed subset of The set is called local unitary spectrum of at We will say that 𝑻 is a multiple of the identity if there is 𝜆 ∈ ℝ such that 𝑇(𝑡) = 𝑒𝑥𝑝(𝑖𝜆𝑡) for all 𝑡 ≥ 0. Theorem. Let be a strongly continuous, one-parameter semi-group of contractions on a Hilbert space 𝐻. Assume that there exists a vector such that and is at most countable. If is not a multiple of the identity, then there exists a nonzero vector such that

We prove that the generalized resolvent operator defined in a Hilbert space cannot remain constant on any open subset of the resolvent set. Under certain conditions we also prove the same result for a complex uniformly convex Banach space. These results extend the known ones.

On the Structure of the Spectrum and the Resolvent Set of a Toeplitz Operator in a Countably Normed Space of Smooth Functions

This paper is devoted to the analysis of the single layer boundary integral operator mathcal {C}_z for the Dirac equation in the two- and three-dimensional situation. The map mathcal {C}_z is the strongly singular integral operator having the integral kernel of the resolvent of the free Dirac operator A_0 and z belongs to the resolvent set of A_0. In the case of smooth boundaries fine mapping properties and a decomposition of mathcal {C}_z in a ‘positive’ and ‘negative’ part are analyzed. The obtained results can be applied in the treatment of Dirac operators with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions that are combined in a critical way.

Abstract We show that if a densely defined closable operator A is such that the resolvent set of A2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that is normal for some , then T is normal. Hence a closed subnormal operator T such that is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that and are self‐adjoint for some coprime numbers p and q, then A must be self‐adjoint.

Abstract Let be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space . Let be the Weyl family corresponding to . We cope with two main topics. First, since need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation , for some , becomes a nontrivial task. Regarding as the (Shmul'yan) transform of induced by Γ, we give conditions for the equality in to hold and we compute the adjoint . As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for is nonempty. Based on the criterion for the closeness of , we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family corresponding to a boundary relation Γ for is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair with its Weyl family . The transformation scheme is either or with suitable linear relations V. Results in this direction include but are not limited to: a 1‐1 correspondence between and ; the formula for , for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple with and (second scheme, Hilbert space case).