In 4 a model is presented of a finite-dimensional Pontryagin space with a symmetric operator without eigenvalues. In this note we show that this model is unique up to an equivalence relation.
For a finite group τ and a field k of characteristic p dividing the order of τ, we construct a map pτ:Uτ→PYτ of varieties over k with unions of affine spaces as fibers and with (PYτ)/τp-isogenous to ...ProjH•(τ,k). On Uτ, we construct a “universal p-nilpotent operator” PΘτ which leads to the construction of τ-equivariant coherent sheaves associated to finite dimensional kτ-modules M. These coherent sheaves are algebraic vector bundles on Uτ if M has constant Jordan type. For any finite dimensional kτ-module M, these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of M.
For a linear algebraic group G, we compare these constructions for the finite group G(Fp) to previous constructions for the infinitesimal group scheme G(r). Our comparison of bundle invariants for a rational G-module M of exponential type <pr upon restriction to G(Fp) and to G(r) is a strengthened form of earlier comparisons of support varieties for M. Our constructions extend to semi-direct products G⋊τ of an infinitesimal group scheme G and a finite group τ, thus to all finite group schemes if k is algebraically closed.
Properties of m-selfadjoint and m-isometric operators have been investigated by several researchers. Particularly interesting to us are algebraic properties of nilpotent perturbations of such ...operators. McCullough and Rodman showed in the nineties that if Qn=0 and A is a selfadjoint operator commuting with Q then the sum A+Q is a (2n−1)-selfadjoint operator. Very recently, Bermúdez, Martinón, and Noda proved a similar result for nilpotent perturbations of isometries. Via a new approach, we obtain simple proofs of these results and other generalizations to operator roots of polynomials.
In this paper, we introduce the notion of
-partial isometry for a positive operator A and a nonnegative integer m. This family of operators contains both the class of
-isometries discussed in Sid ...Ahmed and Saddi A-m-isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436:3930-3942 and that of m-partial isometries introduced in Saddi and Sid Ahmed m-partial isometries on Hilbert spaces. Int J Funct Anal Oper Theory Appl. 2010;2(1):67-83. First, we give some interesting algebraic properties of
-partial isometries, then we discuss a necessary and sufficient condition for an
-partial isometry to be an
-isometry. Finally, we give some spectral properties of
-partial isometries.
For some operator A ? B(H), positive integers m and k, an operator T ? B(H) is called k-quasi-(A,m)-symmetric if T*k( mP j=0 (?1)j(m j )T*m?jATj)Tk = 0, which is a generalization of the m-symmetric ...operator. In this paper, some basic structural properties of k-quasi-(A,m)-symmetric operators are established with the help of operator matrix representation. We also show that if T and Q are commuting operators, T is k-quasi-(A,m)-symmetric and Q is n-nilpotent, then T + Q is (k + n ? 1)-quasi-(A,m + 2n ? 2)-symmetric. In addition, we obtain that every power of k-quasi-(A,m)-symmetric is also k-quasi-(A,m)-symmetric. Finally, some spectral properties of k-quasi-(A,m)-symmetric are investigated.
We prove that if an isometry A and a nilpotent operator Q of order n commute, then A+Q is a strict (2n−1)-isometry. As an application of the main result, we prove that A+Q cannot be N-supercyclic for ...any N. Also, we find an m-isometry with prescribed spectrum K, where K is the closed unit disk or a compact subset of the unit circle.
We say that an operator T ∈ B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: H → H so that T = CT*C. We prove that binormal operators, operators that are ...algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).
A bounded linear operator $T$ on a complex Banach space $\mathcal{X}$ is said to be full if $\overline{T\mathcal{M}}=\mathcal{M}$ for every invariant subspace $\mathcal{M}$ of $\mathcal{X}$. It is ...nearly full if $\overline{T\mathcal{M}}$ has finite codimension in $\mathcal{M}$. In this paper, we focus our attention to characterize full and nearly full operators in complex Banach spaces, showing that some valid results in complex Hilbert spaces can be generalized to this context.
For a positive integer m and a Hilbert space H an operator T in B(H), the space of all bounded linear operators on H, is called m-selfadjoint if ∑k=0m(−1)k(mk)T⁎kTm−k=0. In this paper, we show that ...if T∈B(H) and the spectrum of T consists of a finite number of points then it is m-selfadjoint if and only if it is an n-Jordan operator for some integer n. Moreover, we prove that if T is m-selfadjoint then T is nilpotent when it is quasinilpotent. Then we characterize m-selfadjoint weighted shift operators. Also, we show that if T is m-selfadjoint then so is p(T) when p(z) is a polynomial with real coefficients. After that, we investigate an elementary operator τ and a generalized derivation operator δ on the Hilbert-Schmidt class C2(H) which are m-selfadjoint. Finally, we prove that no m-selfadjoint operator on an infinite-dimensional Hilbert space, can be N-supercyclic, for any N≥1.
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