Extremal compressions of closed operators

Abstract

Abstract.Let X be a Banach space, A a closed linear operator on X , and X\ , ... , A" isolated eigenvalues of A of finite multiplicity. If P is a projection on X such that X\,... ,Xn belong to the resolvent of the compression of A on the range of P it is easy to see that dimN(P) > max{dim N{X¡ -A): I <i <n}.It is shown that there exist such projections where we have equality in this inequality.Let X be a complex Banach space. For an operator A in X (not necessarily everywhere defined) we denote by D(A), N(A), and RiA) the domain, the kernel, and the range of A , respectively. If P is a bounded linear projection on X with PiDiA)) ç DÍA) the compression of A on RÍP) is the linear operator Ap defined in the closed subspace RÍP) by DÍAp) = DÍA) n RÍP) and Apy = PAy for y G RÍP). It is easy to see that Ap is injective (i.e., N{Ap) = {0}) iff NÍA) n RÍP) = {0} and R(AP) n A(P) = {0} . Therefore dim N(P) > dim NÍA) if Ap is injective.The collection of projections Poní with P(DiA)) ç DÍA) will be denoted by PaÍX) . The main result of this note is the following:Theorem. Let A be a closed linear operator in a Banach space X and let Xx, ... , Xn be isolated eigenvalues of A of finite (algebraic) multiplicity. Let Ci be a subset of C such that £2 ^ C ¿i«¿z £2 n (TÍA) = {Xx, ... , Xn\. If C £ C\£2 then there is a projection P £ PaÍX) such that £2 ç píAP), o(Ap) = tr(^)\£2u{C} and dim A(P) = max dim MX -A). A€£2In [ 1, 2] Islamov proved a similar result for finite-dimensional perturbations of A. It is well known that many results on compact or finite-dimensional perturbations have analogues in terms of compressions to subspaces of finite codimension, see [5,6,7]. The theorem above is another example of this relationship.

Structure formation of integral‐asymmetric membranes of polystyrene‐block‐Poly(ethylene oxide)

“…The driving force in the self-assembly in bulk is the microphase separation, which is highly dependent on the molecular weight, Flory-Huggins-Staverman interaction parameter and the volume fraction. [30][31][32] Moreover, BCP show also self-assembly in solution. When the BCP is dissolved in a selective solvent, which is a good solvent for one and a poor solvent for the other block, the last mentioned block will be present in a more collapsed conformation.…”

“…The structure is dependent on the nature of the BCP, temperature, pH and ionic effects, and the selectivity of the solvent. 32 Hence, by combining the conventional phase inversion process and the self-assembly of cylinder forming BCP, membranes comprising of an isoporous thin top layer merging in a porous substructure can be generated, if the cylinders selfassemble perpendicular to the surface of the membrane. So far this method has been successfully demonstrated only for PS-b-P2VP and PS-b-P4VP.…”

“…Polystyrene-block-poly(ethylene oxide) (PS-b-PEO) was synthesized by sequential anionic polymerization as described in the literature before. 25,26 The polymerization of styrene was initiated with sec-BuLi in the presence of the phosphazene base (P 4 -t-Bu) at À70 C. After 2 h ethylene oxide was condensed in the reactor and the solution was stirred for 30 min at À30 C. The temperature of the solution was gradually increased to 40 C and then the polymerization was left to complete for 3 days. The polymerization was quenched with 1% solution of acetic acid in degassed methanol.…”

Triplet harvesting in poly(9‐vinylcarbazole) and poly(9‐(2,3‐epoxypropyl)carbazole) doped with CdSe/ZnS quantum dots encapsulated with 16‐(N‐carbazolyl) hexadecanoic acid ligands

“…The PL spectrum of the PVK film at room temperature shows a peak at 410 nm attributed to the fluorescence of the S 1 (fexcimer) state and a shoulder at 390 nm attributed to the fluorescence of the S 1 (p-excimer) state. In addition, Figure 1(b) shows that the absorption spectrum of the QDs overlaps perfectly with these PL spectra, suggesting the possibility of both singlet (F€ orster [46][47][48] and Dexter 48,49 transfer) from as well the carbazole monomer as the p-and f-excimer and triplet energy transfer (mainly Dexter transfer) to the QDs.…”

“…This means, in the framework of the F€ orster model for energy transfer, that the rate constant of energy transfer for the monomer acting as donor will also be twice as large as for the f-excimer acting as donor, assuming they have a similar fluorescence rate constant; taking into account, the larger fluorescence rate constant of the monomer compared to the f-excimer the effect will be even larger. [46][47][48] Furthermore, the monomer can also transfer energy to the QDs by a sequence of hopping events of the excitation between the carbazoles followed by energy transfer to the QD from a carbazole close to the QD. The more efficient energy transfer by the monomers can, especially for PVK, be observed by the change of the features of the carbazole emission upon increasing the load of QDs [ Fig.…”