Let be a complex unital Banach algebra and be a regularity. In order to characterize the accumulation point of the spectrum of associated with , we define -decomposition of elements in , and give the conditions to make a regularity, where is the set of elements with -decomposition. Further, we study the relation between the spectrum of associated with and the accumulation point of the spectrum of associated with . Besides, some properties of are discussed.

Based on the new concept of 2-tuples of noncommutative Orlicz sequence spaces, we establish the Riesz–Thorin interpolation theorem. As applications, bounds for the nonsquare and von Neumann–Jordan constants of the noncommutative Orlicz space are given.

Let 𝒜 be a unital infinite dimensional semisimple Banach algebra and Φ (𝒜) be the set of Fredholm elements in 𝒜 . An element a ∈𝒜 is called Φ(𝒜) -consistent provided that ab∈Φ (𝒜) if and only if ba∈Φ (𝒜) for every b∈𝒜 . We first characterize the Φ (𝒜) -consistent elements and show that the set of such elements forms an upper semiregularity. Building on this, we introduce the consistent Fredholm spectrum, establish its spectral mapping theorem, and obtain a characterization for algebraic elements in 𝒜 . As an application, we characterize the stability of Φ (𝒜) -consistent elements in terms of nullity and defect in primitive c^* - algebras.

Let A{\mathcal{A}} be a complex unital Banach algebra and (R1,R2{R}_{1},{R}_{2}) be a pair of semiregularities in A{\mathcal{A}}. In order to investigate the boundary of spectra in the axiomatic theory of spectrum, this article defines the concept of spectrum boundary domination (SBD) of semiregularities and gives a criterion for a pair (R1,R2{R}_{1},{R}_{2}) possessing SBD property. Furthermore, the conditions such that σR1(a)=σR2(a){\sigma }_{{R}_{1}}\left(a)={\sigma }_{{R}_{2}}\left(a) are described for a∈Aa\in {\mathcal{A}} under the assumption that the pair (R1,R2{R}_{1},{R}_{2}) possesses SBD property. In addition, the transfer of SBD property through a Banach algebra homomorphism is discussed.

Let G be a finite group. Starting from the field algebra F of G-spin models, we show that the C"-basic construction for the field algebra F and the sic D(G)-invariant subalgebra of F can be represented as the iterated crossed product C"-algebra can be obtained, which is C"-isomorphic to the C"-basic construction for F & rtimes; D(G) and the field algebra F. In addition, it is proved that the iterated crossed product C"-algebra is a new field algebra, and the concrete structures with the order and disorder operators are given.

Let R be an associative ring with unit 1, and let a, b, c∈ R satisfy (ac)^2a=abaca=acaba=a(ba)^2. We prove that if α =1-ba is generalized Drazin invertible, then 1-ac is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson’s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.

The primary contribution of this study lies in proposing a new concept termed 2-tuples of noncommutative Orlicz sequence spaces ⊕_j=1^2S_φ_j,p, where S_φ_j denotes a noncommutative Orlicz sequence space. By leveraging the three-line theorem, we establish the Riesz-Thorin interpolation theorem for ⊕_j=1^2S_φ_j,p. As applications, we derive bound for the nonsquare and von Neumann-Jordan constant of noncommutative Orlicz space S_φ_s (0<s≤1), where φ_s is an intermediate function.

The paper initiates the concept of negative definite consistent in Fredholm. The entanglement stability and the weakly entanglement stability of negative definite consistent in Fredholm for the system of 2× 2 upper triangular operator matrices are characterized with the help of the negative definite consistent in Fredholm spectrum.

A linear bounded operator T on an infinite dimensional separable complex Hilbert space H is said to satisfy property ( UW_E ) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the CI spectrum induced by consistent invertibility property of operators, we explore property ( UW_E ) for T and T^* simultaneously. Furthermore, the transfer of property ( UW_E ) from T to f(T) and f(T^*) is obtained, where f is a function which is analytic in a neighborhood of the spectrum of T. At last, with the help of the so-called (A, B) entanglement stable spectra, the entanglement stability of property ( UW_E ) for 2× 2 upper triangular operator matrices is investigated.

For a given operator pair (A,B)∈ (B(H),B(K)) , we denote by M_C the operator acting on a complex infinite dimensional separable Hilbert space H⊕ K of the form M_C= ( A C 0 B ) . This paper focuses on the Fredholm complement problems of M_C . Namely, via the operator pair (A, B), we look for an operator C∈ B(K,H) such that M_C is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (C) as a variant of Weyl’s theorem. At last, the stability of property (C) for 2× 2 upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (A, B).

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

Let A be a complex unital Banach algebra and let R subset of A be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R-D is constructed as an extension of R to axiomatically study the accumulation of sigma(R)(a) for any a is an element of A. At last, several illustrative examples on Banach algebra and operator algebra are provided.

An operator $T\in B(H)$ is said to satisfy property ($UW${\scriptsize \it{E}}) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the CI spectrum induced by consistent invertibility property of operators, we explore property ($UW${\scriptsize \it{E}}) for $T$ and $T^\ast$ simultaneously. Furthermore, the transfer of property ($UW${\scriptsize \it{E}}) from $T$ to $f(T)$ and $f(T^{\ast})$ is obtained, where $f$ is a function which is analytic in a neighborhood of the spectrum of $T$. At last, with the help of the so-called $(A,B)$ entanglement stable spectra, the entanglement stability of property ($UW${\scriptsize \it{E}}) for $2\times 2$ upper triangular operator matrices is investigated.

An operator T∈ B(H) is said to satisfy property (UWE) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the CI spectrum induced by consistent invertibility property of operators, we explore property (UWE) for T and T^∗ simultaneously. Furthermore, the transfer of property (UWE) from T to f(T) and f(T^∗) is obtained, where f is a function which is analytic in a neighborhood of the spectrum of T. At last, with the help of the so-called (A,B) entanglement stable spectra, the entanglement stability of property (UWE) for 2× 2 upper triangular operator matrices is investigated.

For 1≤ p<∞ , let L_p(ℳ,τ ) be the non-commutative L_p -space associated with a von Neumann algebra ℳ , where ℳ admits a normal semifinite faithful trace τ . Using the trace τ , Banach duality formula and Gâteaux derivative, this paper characterizes an element a∈ L_p(ℳ,τ ) such that ‖ a‖ _p=inf{‖ a+b‖ _p: b∈ℬ_p}, where ℬ_p is a closed linear subspace of L_p(ℳ,τ ) and ‖·‖ _p is the norm on L_p(ℳ,τ ) . Such an a is called ℬ_p -minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces ℬ_p=⊕ _i=1^∞ e_i 𝒮 e_i (converging with respect to ‖·‖ _p ) are considered, where {e_i}_i=1^∞ is a sequence of mutually orthogonal and τ -finite projections in a σ -finite von Neumann algebra ℳ , and 𝒮 is the set of elements in ℳ with τ -finite supports.

Let 𝒜 be a C*-algebra and ℬ a C*-subalgebra of 𝒜 such that there is a conditional expectation from 𝒜 onto it. Using the property of positive modification, this paper characterizes an element a∈𝒜 satisfying ‖ a‖ =inf{‖ a+b‖ : b∈ℬ}. Such an a is called ℬ -minimal. As an application of these results it is shown that both the unilateral shift and the backward shift are D(B(l^2)) -minimal, where D(B(l^2)) is the set of diagonal operators in B(l^2) , and thus provides new examples of minimal operators which are neither hermitian nor compact.

Let H be a finite dimensional C∗-Hopf algebra and A the observable algebra of Hopf spin models. For some coaction of the Drinfeld double D(H) on A, the crossed product A⋊D(H)^ can define the field algebra F of Hopf spin models. In the paper, we study C∗-basic construction for the inclusion A⊆F on Hopf spin models. To achieve this, we define the action α:D(H)×F→F, and then construct the resulting crossed product F⋊D(H), which is isomorphic A⊗End(D(H)^). Furthermore, we prove that the C∗-basic construction for A⊆F is consistent to F⋊D(H), which yields that the C∗-basic constructions for the inclusion A⊆F is independent of the choice of the coaction of D(H) on A.

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A A in M n ( C ) M_n(\Bbb C) such that ‖ A ‖ = min B ∈ B ‖ A + B ‖ , \begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*} where B \mathcal {B} is a C*-subalgebra of M n ( C ) M_n(\Bbb C) , ‖ ⋅ ‖ \|\cdot \| denotes the operator norm. Such an A A is called B \mathcal {B} -minimal. Moreover, for a C*-subalgebra B \mathcal {B} determined by a conditional expectation from M n ( C ) M_n(\Bbb C) onto it, this paper constructs ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal hermitian matrices in M k n ( C ) M_{kn}(\Bbb C) through B \mathcal {B} -minimal hermitian matrices in M n ( C ) M_n(\Bbb C) , and gets a dominated condition that the matrix A ^ = diag ⁡ ( A 1 , A 2 , ⋯ , A k ) \hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots , A_k) is ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal if and only if ‖ A ^ ‖ ≤ ‖ A s ‖ \|\hat {A}\|\leq \|A_s\| for some s ∈ { 1 , 2 , ⋯ , k } s\in \{1,2,\cdots ,k\} and A s A_s is B \mathcal {B} -minimal, where A i ( 1 ≤ i ≤ k ) A_i(1\leq i\leq k) are hermitian matrices in M n ( C ) M_n(\Bbb C) .

In this paper, we extend Jacobson's lemma for Drazin inverses to the generalized \(n\)-strong Drazin inverses in a ring, and prove that \(1-ac\) is generalized \(n\)-strong Drazin invertible if and only if \(1-ba\) is generalized \(n\)-strong Drazin invertible, provided that \(a(ba)^{2}=abaca=acaba=(ac)^{2}a\). In addition, Jacobson's lemma for the left and right Fredholm operators, and furthermore, for consistent in invertibility spectral property and consistent in Fredholm and index spectral property are investigated.

Let $${\mathcal {A}}$$ be a unital semisimple Banach algebra. Denote the set of the generalized Fredholm elements in $${\mathcal {A}}$$ by $$\Phi _{g}({\mathcal {A}})$$ . In this paper, we study the perturbations of the generalized Fredholm elements and the spectral mapping theorem of the generalized Fredholm spectrum. Furthermore, for $$a\in \Phi _{g}({\mathcal {A}})$$ , the conditions that f(a) is also a generalized Fredholm element are investigated, where f is a complex-valued function analytic on a neighborhood of $$\sigma (a)$$ . In addition, the topological structure of $$\Phi _{g}({\mathcal {A}})$$ are discussed. As an application, the socle of a primitive $$C^{*}$$ -algebra is characterized by the generalized Fredholm elements.