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Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem  [PDF]
Michael Frank,Kamran Sharifi
Mathematics , 2007,
Abstract: In this notes unbounded regular operators on Hilbert $C^*$-modules over arbitrary $C^*$-algebras are discussed. A densely defined operator $t$ possesses an adjoint operator if the graph of $t$ is an orthogonal summand. Moreover, for a densely defined operator $t$ the graph of $t$ is orthogonally complemented and the range of $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained. Changes 1: Improved results, corrected misprints, added references. Accepted by J. Operator Theory, August 2007 / Changes 2: Filled gap in the proof of Thm. 3.1, changes in the formulations of Cor. 3.2 and Thm. 3.4, updated references and address of the second author.
On the Fredholm property of bisingular pseudodifferential operators  [PDF]
M. Borsero,J. Seiler
Mathematics , 2014,
Abstract: For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.
Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules  [PDF]
Michael Frank
Mathematics , 2006,
Abstract: B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*-algebras of compact operators can be characterized by the property that every norm-closed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*-module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*-modules over C*-algebras which characterize precisely the C*-algebras of compact operators.
Narrow operators and the Daugavet property for ultraproducts  [PDF]
Dmitriy Bilik,Vladimir Kadets,Roman Shvidkoy,Dirk Werner
Mathematics , 2001,
Abstract: We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or $X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.
On the backward uniqueness property for a class of parabolic operators  [PDF]
Daniele Del Santo,Martino Prizzi
Mathematics , 2006,
Abstract: We give sharp regularity conditions, ensuring the backward uniquess property to a class of parabolic operators.
On operators with bounded approximation property  [PDF]
Oleg Reinov
Mathematics , 2013,
Abstract: It is known that any separable Banach space with BAP is a complemented subspace of a Banach space with a basis. We show that every operator with bounded approximation property, acting from a separable Banach space, can be factored through a Banach space with a basis.
Slice continuity for operators and the Daugavet property for bilinear maps  [PDF]
Enrique A. Sánchez Pérez,Dirk Werner
Mathematics , 2012,
Abstract: We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.
Symbolic calculus and Fredholm property for localization operators  [PDF]
Elena Cordero,Karlheinz Gr?chenig
Mathematics , 2005,
Abstract: We study the composition of time-ffrequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operators possesses the Fredholm property, and thus its range is closed in the target space.
Bari-Markus property for Dirac operators  [PDF]
Ya. V. Mykytyuk,D. V. Puyda
Mathematics , 2014,
Abstract: We prove the Bari-Markus property for spectral projectors of non-self-adjoint Dirac operators on a finite interval with square-integrable matrix-valued potentials and some separated boundary conditions.
A disjointness type property of conditional expectation operators  [PDF]
Beata Randrianantoanina
Mathematics , 2001,
Abstract: We give a characterization of conditional expectation operators through a disjointness type property similar to band preserving operators. We say that the operator $T:X\to X$ on a Banach lattice $X$ is semi band preserving if and only if for all $f, g \in X$, $f \perp Tg$ implies that $Tf \perp Tg$. We prove that when $X$ is a purely atomic Banach lattice, then an operator $T$ on $X$ is a weighted conditional expectation operator if and only if $T$ is semi band preserving.
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