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In this paper, we consider the notions of spherical p, log , and w-hyponormality of commuting pairs of operators and study their properties and relations. We first show that spherically p-hyponormal 2-variable weighted shifts of hyponormal operators are invariant under powers and the generalized spherical Aluthge transformation. Second, we study the relation between spherical p-hyponormality and spherical w-hyponormality. Third, we study whether the spherical p-hyponormality with 0

This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n×nn\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

The goal of this paper is to develop a dilation theory on the regular U -twisted polyball B U (H) , which is the set of all k-tuples T : = (T 1 , … , T k) of row contractions T i : = [ T i , 1 ⋯ T i , n i ] on a Hilbert space H satisfying certain positivity condition on the defect operator Δ T (I) and U -commutation relations, where U ⊂ B (H) is an appropriate set of unitary operators. The role of operator model is played by a standard multi-shift S : = (S 1 , … , S k) with S i : = [ S i , 1 ⋯ S i , n i ] , which is a k-tuple of doubly I ⊗ U -commuting pure row isometries on the Hilbert space ℓ 2 (F n 1 + × ⋯ × F n k +) ⊗ H , where F n i + is the unital free semigroup with n i generators and each S i , j is a weighted shift with operator-valued weights. It is shown that many of the classical results concerning the dilation theory of contractions on Hilbert spaces have analogues for U -twisted polyballs. This includes: Sz.-Nagy dilation theorem, von Neumann inequality, Itô and Brehmer dilations for commuting isometries and contractions, respectively, and Beurling characterization of the invariant subspaces for the unilateral shift on the Hardy space H 2 . [ABSTRACT FROM AUTHOR]

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near S ∗ -invariance property of the image space C φ (ker T) is explored for composition operators C φ , induced by inner functions φ , and Toeplitz operators T. After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels. [ABSTRACT FROM AUTHOR]

In this article, we discuss subspace duals of a frame of translates by an action of a closed abelian subgroup Γ of a locally compact group G. These subspace duals are not required to lie in the space generated by the frame. We characterise translation-generated subspace duals of a frame/Riesz basis involving the Zak transform for the pair (G , Γ). We continue our discussion on the orthogonality of two translation-generated Bessel pairs using the Zak transform, which allows us to explore the dual of super-frames. As an example, we extend our findings to splines, Gabor systems, p-adic fields Q p , locally compact abelian groups using the fiberization map. [ABSTRACT FROM AUTHOR]

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for regular completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation (σ , V) modulo N (V ~) is hyponormal modulo N (V ~) . [ABSTRACT FROM AUTHOR]

Due to the redundancy property of frames, the stable decomposition of a vector in the separable Hilbert space $ \mathcal H $ H allows the flexibility of choosing different types of duals for a frame. For a second countable locally compact group $ \mathscr {G} $ G (not necessarily abelian) and a closed abelian subgroup Γ, we study the properties of oblique Γ-translation generated (Γ-TG) duals for a continuous frame in $ L^2(\mathscr {G}) $ L 2 (G). Two types of oblique Γ-TG duals viz., type-I and type-II are characterized in terms of the Zak transform for the pair $ (\mathscr {G}, \Gamma) $ (G , Γ). Outside the group setup, first, we discuss such duals for the multiplication generated systems on the measure-theoretic abstraction in $ L^2(X; \mathcal {H}) $ L 2 (X ; H) using the range function corresponding to the point-wise conditions in $ \mathcal H $ H , where X is a σ-finite measure space. Our results present a unified theory connecting the discrete problems with a continuous setup. Besides we characterize these duals' uniqueness using the Gramian/dual Gramian operators, which become a discrete frame/Riesz basis for the associated range space. As an application, we illustrate our results for $ \mathbb {R}^n $ R n , p-adic numbers $ \mathbb {Q}_p $ Q p and locally compact abelian groups using fiberization map. [ABSTRACT FROM AUTHOR]

We extend the notion of convex-cyclicity for matrices to that of convex-cyclicity for abelian semigroups of matrices on K n . We say that an abelian semigroup G of matrices on K n is convex-cyclic if there exists x ∈ K n such that the convex hull of the orbit G(x) of x under G is dense in K n . We provide an effective method for checking that a given abelian semigroup is convex-cyclic. In particular we give a spectral characterization of convex-cyclicity for finitely generated abelian semigroups of diagonalizable matrices. We also obtain an example of a convex-cyclic abelian semigroup which does not contain a convex-cyclic matrix. We prove that there exists an abelian semigroup of matrices on K n which is somewhere convex-cyclic but not convex-cyclic. This solves two questions of Rezaei in (Linear Algebra Appl 438:4190–4203, 2013) and Feldman and McGuire in (Oper Matrices 12(2):465–492, 2017) respectively. On the other hand, we provide the link between ε -hypercyclicity and convex-cyclicity for abelian semigroups of matrices on K n for every 0 < ε < 1 . [ABSTRACT FROM AUTHOR]

The invariant subspace problem asks whether every bounded linear operator on a separable complex Hilbert space has a nontrivial invariant subspace. This problem is a long-standing open problem. This note concerns the invariant subspace problem for hyponormal operators. We give three strategies towards an affirmative answer to that problem. [ABSTRACT FROM AUTHOR]

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over N 0 k. As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system. [ABSTRACT FROM AUTHOR]

This paper presents Wold-type decomposition for various pairs of twisted contractions on Hilbert spaces. We achieve an explicit decomposition for pairs of twisted contractions such that the c.n.u. parts of the contractions are in C 00 . The structure for pairs of doubly twisted operators consisting of a power partial isometry has been discussed. It is also shown that for a pair (T , V ∗) of twisted operators with T as a contraction and V as an isometry, there exists a unique (up to unitary equivalence) pair of doubly twisted isometries on the minimal isometric dilation space of T. Finally, we have given a characterization for pairs of doubly twisted isometries. [ABSTRACT FROM AUTHOR]

In this paper, we study the necessary and sufficient conditions to have a solution pair $ (X, Y) $ (X , Y) to the generalized Sylvester equation AX−Y B = C where $ A, B, C\in \mathcal {L}(\mathcal H) $ A , B , C ∈ L (H) are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair $ (X, Y) $ (X , Y) of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators. [ABSTRACT FROM AUTHOR]

The operators T on a Hilbert space H which have 3-isometric liftings S on Hilbert spaces K containing H are investigated. Here, we deal with the liftings S which are 2-isometries on their invariant subspace K ⊖ H and also have this subspace invariant for S ∗ S . Several characterizations for such operators T are obtained, including the case when the lifting S is expansive. As an application, we refer to the class of (A, 2)-contractions for a positive operator A on H , and particularly to the expansive 3-concave operators. Also, we obtain an upper triangulation for T induced by S, where the operators on the main diagonal are close to contractions. Finally, we study such liftings S which are minimal, in the sense that K is generated by { S n H , n ≥ 0 } . [ABSTRACT FROM AUTHOR]

Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its introduction, the notion was extended to some specific classes of non-abelian groups, and a natural problem emerged, i.e., whether it can be extended to the entire class of (including non-abelian) locally compact groups. In this paper we solve this problem. Interestingly enough, the bracket function (or rather an operator in this case) keeps all of its useful properties, except the Cauchy-Schwarz inequality. Nevertheless, we show that cyclic spaces still can be represented in terms of bracket-weighted spaces. [ABSTRACT FROM AUTHOR]

Let X be an infinite-dimensional complex Banach space and let B (X) denote the algebra of all bounded linear operators on X. For an operator T ∈ B (X) the sets σ 1 (T) , σ 2 (T) , and σ 3 (T) are called, respectively, the semi-Fredholm domain, the Fredholm domain, and the Weyl domain, of T in the spectrum, σ (T) . Given i ∈ { 1 , 2 , 3 } , the goal of this article is to describe the general form of all surjective maps ϕ on B (X) which satisfy σ i (ϕ (A) ϕ (T) + ϕ (T) ϕ (A)) = σ i (A T + T A) for all A , T ∈ B (X) . [ABSTRACT FROM AUTHOR]

Consider (T t) t ≥ 0 and (S t) t ≥ 0 as real C 0 -semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup (T t) t ≥ 0 by (S t) t ≥ 0 , which means that - S t v ≤ T t u ≤ S t v holds for all t ≥ 0 and all real u and v that satisfy - v ≤ u ≤ v . This characterisation extends the Ouhabaz characterisation for semigroup domination to the non-commutative L 2 -spaces. Additionally, we present a simpler characterisation when both semigroups are positive as well as consider the setting in which (T t) t ≥ 0 need not be real. [ABSTRACT FROM AUTHOR]

In this paper, we describe all surjective maps ϕ on B (X) , the space of all bounded linear operators on an infinite-dimensional complex Banach space X, which satisfy H 0 (ϕ (T) ϕ (A) - ϕ (A) ϕ (T)) = H 0 (T A - A T) for all A , T ∈ B (X). Similarly for surjective maps ϕ on B (X) satisfying K (ϕ (T) ϕ (A) - ϕ (A) ϕ (T)) = K (T A - A T) for all A , T ∈ B (X). [ABSTRACT FROM AUTHOR]

Let (σ , V (1) , ⋯ , V (k)) be a pure doubly commuting isometric representation of the product system E on a Hilbert space H V. A σ -invariant subspace K is said to be Beurling quotient subspace of H V if there exist a Hilbert space H W , a pure doubly commuting isometric representation (π , W (1) , ⋯ , W (k)) of E on H W and an isometric multi-analytic operator M Θ : H W → H V , such that K = H V ⊖ M Θ H W , where Θ : W H W → H V is an inner operator and W H W is the generating wandering subspace for (π , W (1) , ⋯ , W (k)). In this article, we prove the following characterization of the Beurling quotient subspaces: A subspace K of H V is a Beurling quotient subspace if and only if (I E j ⊗ ((I E i ⊗ P K) - T ~ (i) ∗ T ~ (i))) (t i , j ⊗ I H V) (I E i ⊗ ((I E j ⊗ P K) - T ~ (j) ∗ T ~ (j))) = 0 , where T ~ (i) : = P K V ~ (i) (I E i ⊗ P K) and 1 ≤ i , j ≤ k. As a consequence, we derive a concrete regular dilation theorem for a pure, completely contractive covariant representation (σ , V (1) , ⋯ , V (k)) of E on a Hilbert space H V which satisfies Brehmer–Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz. Nagy–Foias-type factorization of isometric multi-analytic operators and joint invariant subspaces of the compression of the induced representation on the Beurling quotient subspace. [ABSTRACT FROM AUTHOR]

The bounded linear operators T on a Hilbert space which have 2-isometric liftings S on are investigated in this paper. We refer to the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which is invariant to the orthogonal projection onto the kernel of S∗S − I. In this special case we describe the canonical triangulation of T induced by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of T and of its adjoint T∗ obtained by means of asymptotic limits associated to T∗ and T, relative to a lifting S for T. The canonical triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is obtained. [ABSTRACT FROM AUTHOR]

Let S n denote the unit sphere in C n . Let H 2 (S n) be the standard Hardy space given by the closure in L 2 (S n) of the polynomials in the coordinate functions of z 1 , ⋯ , z n . For 2-tuple { T 1 , T 2 } acting on H 2 (S 2) , we prove that A ′ (T j) ≃ u ⨁ 1 ∞ M z , moreover, K 0 (A ′ (T jk)) ≅ Z (j = 1 , 2 ; k = 0 , 1 , ⋯) . Furthermore, the commutant and strong irreducibility corresponding to the 2-tuple are discussed. [ABSTRACT FROM AUTHOR]

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples. [ABSTRACT FROM AUTHOR]

We obtain a factorization of the characteristic function of a contractive two-step iterated lifting in terms of the characteristic functions of constituent liftings of the iterated lifting and the Julia–Halmos matrix. We also give an expression for the characteristic function of the minimal part of a contractive two-step iterated lifting as a restriction of the product of the characteristic functions of constituent liftings of the iterated lifting. [ABSTRACT FROM AUTHOR]

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

In this article, we introduce a new class of conjugations in the scalar-valued Hardy space H C 2 (D) and provide a characterization of a complex symmetric Toeplitz operator T ϕ with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator T Φ on the vector-valued Hardy space H C 2 2 (D) with respect to certain conjugations introduced in [Câmara MC, Kliś-Garlicka K, Ptak M. Complex symmetric completions of partial operator matrices. Linear and Multilinear Algebra. 2019; DOI: 10.1080/03081087.2019.1631246], [Kang D, Ko E, Lee JE. Remarks on complex symmetric Toeplitz operators. Linear Multilinear Algebra. 2020; DOI: 10.1080/03081087.2020.1842847], [Ko E, Lee JE. Remark on complex symmetric operator matrices. Linear Multilinear Algebra. 2019;67(6):1198–1216]. [ABSTRACT FROM AUTHOR]

The class of m-isometries on a Hilbert space which admit Brownian unitary extensions is investigated. Brownian unitaries in this context for integers m ≥ 3 , naturally appear as a generalization of the same concept defined by Agler-Stankus for 2-isometries. By contrast with the case m = 2 , here just certain expansive m-isometries have Brownian unitary extensions, when m ≥ 3 . In this context we describe the m-Brownian unitaries, as well as the operators which have such extensions, these being called sub-Brownian m-isometries. Also we characterize the operators which have sub-Brownian m-isometric liftings, obtained by the coupling of an isometry. We refer here, in particular, to the operators similar to compressions of sub-Brownian (m - 1) -isometries. As examples we describe some weighted shifts with scalar weights, which are sub-Brownian m-isometries. Our description is given in terms of the polynomial which determine the weights of such an operator. As an application we prove that the operators with polynomial growth conditions of the powers have m-Brownian unitary dilations. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any infinite-dimensional non-Archimedean Banach space of countable type admits a spectral quasinilpotent operator without a nontrivial closed invariant subspace. [ABSTRACT FROM AUTHOR]

In this paper we study large classes of noncommutative varieties included in admissible noncommutative polydomains in B (H) n , where B (H) is the algebra of all bounded linear operators on a Hilbert space H , in connection with their universal operator models and the corresponding noncommutative Hardy algebras and C ∗ -algebras they generate. The main goal of the paper is to develop a free analogue of the Sz.-Nagy–Foiaş theory of contractions for admissible noncommutative varieties. [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a complex Hilbert space ℋ is called n-normal if T*Tn = TnT*. By Fuglede's theorem T is n-normal if and only if Tn is normal. Let k, n ∈ ℕ. Then a bounded linear operator T is said to be of type I k-quasi-n-normal if T*k{T*Tn − TnT*}Tk =0, and T is said to be of type II k-quasi-n-normal if T*k{T*nTn − TnT*n}Tk =0. 1-quasi-n-normal is called quasi-n-normal. We shall show that (1) type I quasi-2-normal and type II quasi-2-normal are different classes; (2) the intersection of the class of type I quasi-2-normal and the class of type II quasi-2-normal is equal to the class of 2-normal. We also give some examples of type I k-quasi-n-normal and type II k-quasi-n-normal. We also show that Weyl's theorem holds for this class of operators and every k-quasi-n-normal operator has a non trivial invariant subspace. [ABSTRACT FROM AUTHOR]

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent. [ABSTRACT FROM AUTHOR]

Let L (X) be the algebra of all bounded linear operators on an infinite-dimensional Banach space X. Let λ 0 be a fixed complex scalar. Let X T ({ λ 0 }) denote the local spectral subspace of an operator T ∈ L (X) associated with { λ 0 } . The purpose of this paper is to characterize maps ϕ on L (X) that satisfy X ϕ (T) ϕ (S) ({ λ 0 }) = X TS ({ λ 0 }) for all T , S ∈ L (X). We also characterize maps ϕ on L (X) for which the range contains all operators of rank at most four and X ϕ (T) ϕ (S) ϕ (T) ({ λ 0 }) = X TST ({ λ 0 }) for all T , S ∈ L (X). [ABSTRACT FROM AUTHOR]

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators. [ABSTRACT FROM AUTHOR]

For a unitary matrix X of order n over the field of complex numbers and an entire function φ belonging to the Fock space F 2 : = F 2 (C n) , we define an integral operator on F 2 (C n) of the form (H φ X f) (z) = ∫ C n f (w) φ (z + X ∗ Xw ¯) e z w ¯ d λ (w). Here d λ (z) = π - n e - | z | 2 d z is a Gaussian measure on C n . We characterize all the symbols φ for which the operator H φ X is bounded. Next, we consider integral operator on F 2 defined by (R φ f) (z) = ∫ C n f (w) φ (z ⋆ w ¯) d λ (w) for φ ∈ F 2 , where ⋆ is a coordinatewise multiplication. We give a complete characterization for the symbols φ ∈ F 2 (C n) so that the operator R φ is bounded on F 2 . In addition to boundedness, we also obtain some fundamental results for the operators H φ X and R φ such as normality, the C ∗ -algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections B X = { H φ X ∈ B (F 2) : φ ∈ F 2 } , R = { R φ ∈ B (F 2) : φ ∈ F 2 } , respectively. [ABSTRACT FROM AUTHOR]

A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra. [ABSTRACT FROM AUTHOR]

A commuting triple of Hilbert space operators (A, B, P) for which the closure of the tetrablock E , where E = { (x 1 , x 2 , x 3) ∈ C 3 : | x 3 | < 1 & x 1 = c 1 + x 3 c ¯ 2 , x 2 = c 2 + x 3 c ¯ 1 , c 1 , c 2 ∈ C with | c 1 | + | c 2 | < 1 } , is a spectral set is called a tetrablock contraction or an E -contraction. To every E -contraction (A, B, P), there are unique operators F 1 , F 2 ∈ B (D P) , which are called the fundamental operators of (A, B, P), satisfying A - B ∗ P = D P F 1 D P , B - A ∗ P = D P F 2 D P. An E -contraction (A, B, P) admits a canonical decomposition (A 1 ⊕ A 2 , B 1 ⊕ B 2 , P 1 ⊕ P 2) into an E -unitary (A 1 , B 1 , P 1) and a completely non-unitary (c.n.u.) E -contraction (A 2 , B 2 , P 2) . As there already exists an easily understood model for an E -unitary in the literature, it suffices to restrict attention to the c.n.u. E -contraction part. Here we construct an explicit minimal E -isometric dilation for a c.n.u. E -contraction whose fundamental operators satisfy (1.1) below (a class for which it is known that such dilation exists). As a consequence of this explicit dilation, we obtain a functional model for the same class of E -contractions. With the help of this functional model we express an E -contraction (A, B, P) as A = C 1 + P C 2 ∗ , B = C 2 + P C 1 ∗ for some operators C 1 , C 2 and this representation is operator theoretic analogue of the representation of the points in E . We also construct a different functional model, which is not necessarily commutative, for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . This functional model is obtained even without having an E -isometric dilation exhibited in the model. We show by an example that such a model may not be possessed by (A, B, P) if the condition that A, B commute with P ∗ , is dropped from the hypothesis. A complete unitary invariant is achieved for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . The fundamental operators play the central role in all these constructions. Also, we produce a new characterization for an E -unitary. [ABSTRACT FROM AUTHOR]

Let L (X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let XT({λ}) denote the local spectral subspace of an operator T ∈ L (X) associated with a singleton { λ } ⊂ C , and fix a scalar λ 0 ∈ C . We characterize maps ϕ on L (X) which satisfy X ϕ (T) − ϕ (S) ({ λ 0 }) = X T − S ({ λ 0 }) for all T , S ∈ L (X). Furthermore, we give a characterization of maps ϕ on L (X) for which ϕ(0) = 0 and X ϕ (T) + ϕ (S) ({ λ 0 }) = X T + S ({ λ 0 }) for all T , S ∈ L (X). [ABSTRACT FROM AUTHOR]

In this study, we consider two classes of Toeplitz operators on H p , 1 < p < ∞ : Toeplitz operators with unimodular symbols and Toeplitz operators whose spectra satisfy a specific geometric condition (the circular convexity condition). We give some inclusions for their spectrum and some estimates for their resolvents. Using obtained results, we show the existence of nontrivial invariant subspaces of these types of Toeplitz operators. This result gives a partially answer to the question of which type operators on a Banach space has a nontrivial invariant subspace. [ABSTRACT FROM AUTHOR]

In this paper, we study the structures of the invariant subspaces under the action of orthogonal group O2ν(Fq,S){O}_{2\nu }\left({F}_{q},S). In particular, we give a detailed description of 2-codimensional invariant subspaces. Moreover, we show that the height of transfer ideal Im(TrO2ν(Fq,S)){\rm{Im}}({{\rm{Tr}}}^{{O}_{2\nu }\left({F}_{q},S)}) is 2 and give a primary decomposition for the radical ideal of this transfer ideal.