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In this paper, using some properties about Toeplitz kernels, we present some results about finite-rank properties of the commutator [ A f , A g ] . Firstly, we show that [ A B n , A v ∗ ] must have a finite rank on the model space K u 2 , where B n is a finite Blaschke product and v is an inner function. Next, we present that when ker T u ¯ B n is an invariant subspace of T ϕ ∗ , then [ A B n , A ϕ ∗ ] has a finite rank on K u 2 for ϕ ∈ H ∞ . Finally, we prove that [ A B n , A ϕ ∗ ] must have a finite rank on K u 2 when u = B n u 1 for an inner function u 1 . [ABSTRACT FROM AUTHOR]

Let f (z) = def ∑ m = 0 ∞ c m z m be an analytic function with positive Taylor coefficients c m ⩾ 0 , with R f > 0 as its radius of convergence, let T , S : R → C be trigonometrical polynomials and f T S , t , f T ¯ T , t , f S ¯ S , t their associated analytic functions. Let also { A n } n = 1 ∞ , { B n } n = 1 ∞ , { C n } n = 1 ∞ and { D n } n = 1 ∞ be strongly square summable families in each of them consisting of mutually commuting normal operators, satisfying also that A m C n = C n A m and B m D n = D n B m for all m , n ∈ N. Then for any symmetrically norming function Φ , for any and t ∈ R , such that 1 | | | ∑ n = 1 ∞ (A n ∗ A n - C n ∗ C n ) | 1 2 ( f T S , t ( ∑ n = 1 ∞ A n ⊗ B n ) X - f T S , t ( ∑ n = 1 ∞ C n ⊗ D n ) X ) × | ∑ n = 1 ∞ (B n B n ∗ - D n D n ∗) | 1 2 | | Φ ⩽ | | | f T ¯ T , t ( ∑ n = 1 ∞ A n ∗ A n ) - f T ¯ T , t ( ∑ n = 1 ∞ C n ∗ C n ) | 1 2 ∑ n = 1 ∞ (A n X B n - C n X D n ) × | f S ¯ S , t ( ∑ n = 1 ∞ B n B n ∗ ) - f S ¯ S , t ( ∑ n = 1 ∞ D n D n ∗ ) | 1 2 | | Φ , if max { | | ∑ n = 1 ∞ A n ∗ A n | | , | | ∑ n = 1 ∞ B n B n ∗ | | , | | ∑ n = 1 ∞ C n ∗ C n | | , | | ∑ n = 1 ∞ D n D n ∗ | | } < R f. We also provide versions of the inequality (1) for Q, Q ∗ and the Schatten-von Neumann ideals of compact operators, with reduced requirements for the normality and commutativity for the observed families, if ∑ n = 1 ∞ (A n A n ∗ - C n C n ∗) or ∑ n = 1 ∞ (B n ∗ B n - D n ∗ D n) is a semi-definite operator. [ABSTRACT FROM AUTHOR]

We introduce a new class of fuzzy control functions and define the concept of matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability. Then, we apply Radu–Mihet method derived from an alternative fixed point theorem to investigate existence, uniqueness and matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability of a a class of P -Hilfer fractional differential equations in matrix-valued fuzzy Banach space. Next, we show the main results for unbounded domains. An example is given to illustrate the Mittag–Leffler–Hypergeometric–Wright stability for a fractional system. [ABSTRACT FROM AUTHOR]

Let ℋ{\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ(ℋ){\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ{\mathcal{ {\mathcal H} }}. In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators (A,B)∈ℬ(ℋ)×ℬ(ℋ)\left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥AX−XB−I∥≥1,for allX∈ℬ(ℋ).\parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

Applying the Radu-Miheţ method derived from an alternative FPT, we stabilize the tri-additive Υ-random operator inequality by providing a class of stochastic matrix control functions in matrix MB-algebras. In addition, we get an estimation for permuting tri-homomorphisms and permuting tri-derivations in unital matrix MC-⋄-algebras and matrix MB-algebras. [ABSTRACT FROM AUTHOR]

Let U be a unital algebra over a unital commutative ring R and M be a U -bimodule. A trivial extension algebra U ⋉ M is defined as an R -algebra with the usual operations of the R -linear space U ⊕ M and the multiplication defined by (u 1 , m 1) (u 2 , m 2) = (u 1 u 2 , u 1 m 2 + m 1 u 2) for all u 1 , u 2 ∈ U , m 1 , m 2 ∈ M. In this article, the structure of Lie triple derivations on U ⋉ M are given, and it is shown that under some mild restrictions every Lie triple derivation on U ⋉ M is of the form δ + τ , where δ : U ⋉ M → U ⋉ M is a derivation and τ : U ⋉ M → Z (U ⋉ M) is an R -linear mapping. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]

A Hilbert space operator A ∈ B (H) is left (X , m) -invertible by B ∈ B (H) (resp., B ∈ B (H) is an (X , m) -adjoint of A ∈ B (H)) for some operator X ∈ B (H) if △ B , A m (X) = ∑ j = 0 m (− 1) j ( m j ) B m − j X A m − j = 0 (resp., δ B , A m (X) = ∑ j = 0 m (− 1) j ( m j ) B (m − j) X A j = 0). No Drazin invertible operator A ∈ B (H) , with Drazin inverse A d , can be left (I , m) -invertible (equivalently, m-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin invertible operators A, it is seen that the existence of an X acts as a conduit for implications △ B , A (X) = 0 ⟹ δ C , A m (X) = 0 , where the pair i s either (A , A d) or (A d , A) or (A ∗ , A d ∗) or (A d ∗ , A ∗). Reverse implications fail. Assuming certain commutativity conditions, it is seen that △ A d ∗ , A m (X) = 0 = △ B d ∗ , B n (Y) implies δ A ∗ B ∗ , A B m + n − 1 (X Y) = 0 = δ A ∗ + B ∗ , A + B m + n − 1 (X Y). [ABSTRACT FROM AUTHOR]

Let u be an inner function and K u be the corresponding model space. In this article, we characterize the compact commutator [ A f , A g ] : = A f A g - A g A f of truncated Toeplitz operators A f and A g on K u. For f and g inner functions, we discuss the relationship between closed singular set σ (u) and the compactness of [ A f , A g ∗ ]. In particular, when u = f u 1 for inner function u 1 , we present necessary and sufficient conditions for [ A f , A f ∗ ] to have finite rank or to be compact on K u. For f , g ∈ H ∞ , in view of symbols of truncated Toeplitz operators, we show a sufficient condition for [ A f , A g ∗ ] to be compact. For f , g ∈ L ∞ , we obtain necessary and sufficient conditions for [ A f , A g ] to be compact using the function algebra. [ABSTRACT FROM AUTHOR]

Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$. Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$. It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$. We also propose some related topics for future research. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space with dim X ≥ 2 , and A ⊆ B (X) be a standard operator algebra. We show that a linear mapping δ : A → A is anti-derivable at zero (i.e., x y = 0 in A implies y δ (x) + δ (y) x = 0 ) if and only if δ = 0 . [ABSTRACT FROM AUTHOR]

Let ℒ = − Δ + V be a Schrödinger operator, where Δ is the Laplacian operator on ℝd (d ≥ 3), while the nonnegative potential V belongs to the reverse Hölder class Bq, q > d/2. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón—Zygmund operators and Littlewood—Paley functions. These results substantially generalize some well-known results. [ABSTRACT FROM AUTHOR]

Suppose R is a 2,3-torsion free unital alternative ring having an idempotent element e 1 e 2 = 1 - e 1 which satisfies x R · e i = { 0 } ⇒ x = 0 i = 1 , 2 . In this paper, we aim to characterize the commuting maps. Let φ be a commuting map of R so it is shown that φ (x) = z x + Ξ (x) for all x ∈ R , where z ∈ Z (R) and Ξ is an additive map from R into Z (R) . As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative C ∗ -algebra with no central summands of type I 1 . [ABSTRACT FROM AUTHOR]

Let H be an infinite dimensional Hilbert space and B (H) be the C ∗ -algebra of all bounded linear operators on H , equipped with the operator-norm. By improving the Brown–Pearcy construction, Tao (J. Oper. Theory82(2) (2019) 369–382) extended the result of Popa (On commutators in properly infinite W ∗ -algebras, in: Invariant subspaces and other topics (1982) (Boston, Mass.: Birkhäuser, Basel)) which reads as: for each 0 < ε ≤ 1 / 2 , there exist D , X ∈ B (H) with ‖ [ D , X ] - 1 B (H) ‖ ≤ ε such that ‖ D ‖ ‖ X ‖ = O log 5 1 ε , where [ D , X ] : = D X - X D . In this paper, we show that Tao’s result still holds for certain class of unital C*-algebras which include B (H) as well as the Cuntz algebra O 2 . [ABSTRACT FROM AUTHOR]

In this paper we present how to apply a Berberian’s technique to asymmetric Putnam–Fuglede theorems. In particular, we proved that if A , B ∈ B (H) belong to the union of classes of ∗ -paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and A X = X B ∗ for some X ∈ B (H) , then A ∗ X = X B . Moreover, we gave a new counterexample for an asymmetric Putnam–Fuglede theorem for paranormal operators. [ABSTRACT FROM AUTHOR]

A new class of operators, larger than ∗\ast -finite operators, named generalized ∗\ast -finite operators and noted by Gℱ∗(ℋ){{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ∗(ℋ)={(A,B)∈ℬ(ℋ)×ℬ(ℋ):∥TA−BT∗−λI∥≥∣λ∣,∀λ∈C,∀T∈ℬ(ℋ)}.{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.

The characterization of the points in Cp:1≤p

We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten pp-norm (1≤p

Let T be a triangular n-matrix ring ( n ≥ 2). It is shown that, under some mild assumptions, a map δ : T → T is a multiplicative Lie derivation if and only if δ (X) = d (X) + γ (X) holds for all X ∈ T , where d : T → T is an additive derivation and γ : T → Z (T) is a central-valued map vanishing on all commutators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the notion of quasi-multipliers to the general topological algebra setting, not necessarily normed or locally convex. We discuss the quasi-uniform and quasi-strong operator topologies on the algebra QM(A) of all bilinear and jointly continuous quasi-multipliers on A and study their various properties. [ABSTRACT FROM AUTHOR]

The question whether every operator on infinite-dimensional Hilbert space 퐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 푇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 푋 has a nontrivial invariant subspace. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hyers-Ulam-Rassias stability of (m,n)(m,n)-Jordan derivations. As applications, we characterize (m,n)(m,n)-Jordan derivations on C⁎{C}^{\ast }-algebras and some non-self-adjoint operator algebras.

This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann's work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are categorically central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles. In this second article (Part II), we show that categorical Galois theory applies to the inclusion of the category of coverings into the category of surjective morphisms of racks or of quandles. We characterise the induced Galois-theoretic concepts of trivial covering, normal covering and covering in this two-dimensional context. The latter is described via our definition and study of double coverings, also called algebraically central double extensions of racks (and quandles). We define a suitable and well-behaved commutator which captures the zero-, one- and two-dimensional concepts of centralization in the category of quandles. We keep track of the links with the corresponding concepts in the category of groups. [ABSTRACT FROM AUTHOR]

Let A be a unital algebra and M be a unital A -bimodule. We characterize the linear mappings δ and τ from A into M , satisfying δ (A) B + A τ (B) = 0 for every A , B ∈ A with A B = 0 when A contains a separating ideal T of M , which is in the algebra generated by all idempotents in A . We apply the result to P -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that A is a unital Banach algebra and M is a unital Banach A -bimodule, we give a complete description of linear mappings δ and τ from A into M , satisfying δ (A) B + A τ (B) = 0 for every A , B ∈ A with A B = I . [ABSTRACT FROM AUTHOR]

Let M be a finite von Neumann algebra with no central summands of type I 1 . Assume that δ : M → M is a nonlinear map satisfying δ ([ [ A , B ] , C ]) = [ [ δ (A) , B ] , C ] + [ [ A , δ (B) ] , C ] + [ [ A , B ] , δ (C) ] for any A , B , C ∈ M with A B C = 0 . Then, we prove that there exists an additive derivation d : M → M , such that δ (A) = d (A) + τ (A) for any A ∈ M , where τ : M → Z M is a nonlinear map, such that τ ([ [ A , B ] , C ]) = 0 for any A , B , C ∈ M with A B C = 0 . [ABSTRACT FROM AUTHOR]

Let T be a strongly singular Calderón-Zygmund operator and b ∈ Lloc(ℝn). This article finds out a class of non-trivial subspaces BMOω,p,n(ℝn)of BMO(ℝn) for certain ω ∈ A1, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator [b, T] is bounded from weighted Hardy space H ω p (ℝ n) to weighted Lebesgue space L ω p (ℝ n) if b ∈ BMOω,p,∞(ℝn), and is bounded from weighted Hardy space H ω p (ℝ n) to itself if T*1 = 0 and b ∈ BMOω,p,u(ℝn)for 1 < u < 2. [ABSTRACT FROM AUTHOR]

The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

Let σ n (H) denote the class of operators A acting on a complex Hilbert space H such that W 0 (A) = c o (λ ∈ σ (A) : λ = A ) , where W 0 (A) and σ (A) are the maximal numericl range, the spectrum of A, respectively, and co stands for the convex hull. In this paper, we show that the class σ n (H) contains hyponormal operators. We also show that if two operators A ∈ σ n (H) and B ∈ σ n (H ′) have the same closure of the numerical range or have the same spectrum, then they also have the same maximal numerical range. An affirmative answer to the question 2.16 in Fialkow [Elementary operators and applications. Proceedings of the International Workshop; Bleuberen; 1991. p. 55–113] is also given. [ABSTRACT FROM AUTHOR]

Let A be a von Neumann algebra, a linear map δ : A → A is said to be derivable at Ω ∈ A , if δ (A B) = δ (A) B + A δ (B) for every A , B ∈ A with A B = Ω. In this paper, we show that a norm-continuous linear map δ : A → A is derivable at an arbitrary but fixed operator Ω ∈ A if and only if there exists a linear derivation τ : A → A such that δ (A) = τ (A) + δ (I) A for all A ∈ A , where δ (I) is in the centre of A and δ (I) Ω = 0. When A is a von Neumann algebra with no summands of type I 1 or a properly infinite von Neumann algebra, we obtain the same result by weakening the linearity and norm-continuity assumption of δ into additivity. [ABSTRACT FROM AUTHOR]

We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of compactness of Hankel operators acting between distinct Hardy spaces generated by Banach function lattices. We prove an analogue of Hartman’s theorem characterizing compact Hankel operators in terms of properties of their symbols. As a byproduct we give an estimation of the essential norm of such operators. Furthermore, compactness of commutators and semicommutators of Toeplitz operators for unbounded symbols is discussed. [ABSTRACT FROM AUTHOR]

We prove that for normal operators N1, N 2 ∈ ℒ (ℋ) , the generalized commutator [N1, N2; X] approaches zero when [N1, N2; [N1, N2; X]] tends to zero in the norm of the Schatten-von Neumann class C p with p > 1 and X varies in a bounded set of such a class. [ABSTRACT FROM AUTHOR]

Let E,F,G,D be infinite complex Banach spaces and B (F , E) the Banach space of all bounded linear operators from F into E. Consider A 1 , A 2 ∈ B (F , E) , B 1 , B 2 ∈ B (D , G). Let M A 1 , B 1 : X → A 1 X B 1 be the multiplication operator on B (G , F) induced by A 1 , B 1 . In particular, L A 1 = M A 1 , I and R B 1 = M I , B 1 , where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B (G , F) is the sum of two multiplication operators Ψ = M A 1 , B 1 + M A 2 , B 2 . This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations M A 1 , B 1 (X) = C 1 and M A 2 , B 2 (X) = C 2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation M A , B (X) = C. As a consequence, we obtain well-known results of Daji c ´ and Koliha. [ABSTRACT FROM AUTHOR]

We study matrix weights defined on the multivariate torus T d . Sufficient conditions for a matrix weight to be in the Muckenhoupt A 2 -class are studied, and two such sufficiency results obtained by S. Bloom for d = 1 are generalized to the multivariate setting. As an application, an A 2 -decomposition property is introduced for matrix weights and a BMO distance theorem for matrix weights is considered. [ABSTRACT FROM AUTHOR]

2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra M 2 n (A) ( n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property. [ABSTRACT FROM AUTHOR]

Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I. Let p n (X 1 , X 2 , ⋯ , X n) be the polynomial defined by n indeterminates X 1 , ⋯ , X n and their multiple *-Lie products and N be the set of non-negative integers. In this paper, it is shown that if A is closed under the adjoint operation and D = { d m } m ∈ N is the family of mappings d m : A → B (H) such that d 0 = i d A , the identity map on A satisfying d m (p n (U 1 , U 2 , ⋯ , U n)) = ∑ i 1 + i 2 + ⋯ + i n = m p n (d i 1 (U 1) , d i 2 (U 2) , ⋯ , d i n (U n)) for all U 1 , U 2 , ⋯ , U n ∈ A and for each m ∈ N , then D = { d m } m ∈ N is an additive *-higher derivation. Moreover, D is inner. [ABSTRACT FROM AUTHOR]

Let A be a ring and σ : A → A a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map d : A → A for which there exists a map δ : A → A such that d (x y) = δ (x) y + σ (x) d (y) for all x , y ∈ A. If A is a prime C ∗ -algebra and σ is surjective, we determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators E ℓ (A) on A; all such maps are of the form d (x) = b x + a x c for suitable elements a,b,c of the multiplier algebra M (A). As a consequence, if an epimorphism σ : A → A lies in the cb-norm closure of E ℓ (A) , then σ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime C ∗ -algebras like C (X , M 2). [ABSTRACT FROM AUTHOR]

In this paper, the authors first prove the characterizations of Herz Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Herz Triebel-Lizorkin spaces, the author show that the commutators of sublinear operators with Lipschitz functions are also bounded from Herz spaces to Herz Triebel-Lizorkin spaces. [ABSTRACT FROM AUTHOR]