Let H be a complex Hilbert space and let (H) be the algebra of all bounded linear operators on H. The polar decomposition theorem asserts that every operator Te90(H) can be uniquely written as TT, where [T] (TT) and is a partial isometry, called the polar factor of T, whose kernel coincides with that of T. In this paper, we obtain various representations and approximations of the polar factors of operators. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

Let M-n(C) be the algebra of all n & times; n-complex matrices, and similar to(u) be the unitary similarity relation on M-n(C). For any matrix T is an element of M-n(C), let T- be its complex conjugate and |T| := ((TT)-T-& lowast;) (1/ 2) be its absolute value. If T := V-T |T| is its canonical polar decomposition, then its mean transform is M(T) := 1/2 (V-T |T| + |T|V-T) . In this paper, we prove that a bijective map Phi on M-n(C) satisfies M (Phi(T)Phi(S)) similar to(u) Phi(M(TS)) , (T,S is an element of M-n(C)), if and only if there exists a unitary matrix U is an element of M-n(C) such that either Phi(T)= UTU(& lowast; )for all T is an element of M-n(C), or Phi(T)= UTU & lowast; for all T is an element of M-n(C). As a variant of this result, we obtain a complete characterization of surjective maps Phi on M-n(C) satisfying M(Phi(T)Phi(S)) similar to(u) M(TS) for all T, S is an element of M-n(C). (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let B(H) be the algebra of all bounded linear operators acting on a complex Hilbert space H. The polar decomposition theorem asserts that every operator T is an element of B(H) can be uniquely written as T = V-T |T|, the product of a partial isometry V-T is an element of B(H)that has the same kernel as that of T and the modulus |T| := (T & lowast;T)(1/2) of T. In this paper, we obtain the form of all bijective linear maps Phi on B(H)for which V-Phi(T) and V-Phi(S) are unitary similar whenever T, S is an element of B(H) are two operators unitary similar. We also obtain the form of all bijective linear maps on B(H) for which Phi(V-T ) = V-Phi(T) for all T is an element of B(H). Furthermore, a number of related results and consequences is obtained

Let ℬ(H) be the algebra of all bounded linear operators on a complex Hilbert space H, and 𝒞ℛ(H) be the set of all operators in ℬ(H) with closed range. For any operator A∈ℬ(H) , let A^* and A^† be the adjoint and Moore–Penrose inverse of A, respectively. Fix a positive scalar α , and let Θ _α be the transformation defined on 𝒞ℛ(H) by Θ _α (A):= 1/α +1( α A + A^*†) , (A∈𝒞ℛ(H)). In this paper, we characterize all maps Φ on ℬ(H) preserving the closedness of the ranges of the difference of operators and for which Θ _α (Φ (A)-Φ (B)) and Θ _α( A-B) are equivalent by unitaries for all A, B∈ℬ(H) . Then we use such a characterization to obtain the form of all bijective linear maps Φ on ℬ(H) for which Θ _α( Φ (A)) and Θ _α( Φ (B)) are equivalent by unitaries whenever so are A, B∈ℬ(H) . Moreover, we obtain similar results when the relation equivalence by unitaries is replaced by unitarily similarity. Furthermore, a number of related results and consequences is obtained.

Let $ \mathcal {A} $ A and $ \mathcal {B} $ B be two unital C*-algebras with units $ e_\mathcal {A} $ eA and $ e_\mathcal {B} $ eB respectively, and let $ {\bf {V}}({\bf {a}}) $ V(a) and $ {\bf {v}}({\bf {a}}) $ v(a) be the joint numerical range and joint numerical radius, respectively, of any n-tuple $ {\bf {a}}=(a_1, \ldots, a_n) $ a=(a1,& mldr;,an) in $ \mathcal {A}<^>n $ An. Let $ {\bf {e}}_\mathcal {A}=(e_\mathcal {A}, \ldots, e_\mathcal {A}) $ eA=(eA,& mldr;,eA) and $ {\bf {e}}_{\mathcal {A},j} $ eA,j be the n-tuple in $ \mathcal {A}<^>n $ An whose $ j<^>{{\rm th}} $ jth entry is $ e_\mathcal {A} $ eA and all other entries are 0. In this paper, we provide a complete characterization of all surjective mappings $ \Phi : \mathcal {A}<^>n\longrightarrow \mathcal {B}<^>n $ Phi:An -> Bn satisfying $ {\bf {v}}(\Phi ({\bf {a}})-\Phi ({\bf {b}}))= {\bf {v}}({\bf {a}}-{\bf {b}}) $ v(Phi(a)-Phi(b))=v(a-b) for all $ {\bf {a}}, {\bf {b}}\in \mathcal {A}<^>n $ a,b is an element of An under the mild condition $ \Phi ({\bf {e}}_{\mathcal {A},j})-\Phi ({\bf {0}}) $ Phi(eA,j)-Phi(0) is in the centre of $ \mathcal {B}<^>n $ Bn for any $ 1\le j\le n $ 1 <= j <= n. As a consequence, we conclude that a surjective map Phi from $ \mathcal {A}<^>n $ An onto $ \mathcal {B}<^>n $ Bn satisfies $ {\bf {V}}(\Phi ({\bf {a}})-\Phi ({\bf {b}}))= {\bf {V}}({\bf {a}}-{\bf {b}}) $ V(Phi(a)-Phi(b))=V(a-b) for all $ {\bf {a}}, {\bf {b}}\in \mathcal {A}<^>n $ a,b is an element of An if and only if there is a linear Jordan *-isomorphism $ \psi : \mathcal {A}\longrightarrow \mathcal {B} $ psi:A -> B such that $$\begin{align*} \displaystyle\Phi({\bf{x}})=\left(\psi(x_1), \ldots, \psi(x_n)\right)+\Phi({\bf{0}}) \end{align*}$$Phi(x)=(psi(x1),& mldr;,psi(xn))+Phi(0) for all $ {\bf {x}}=(x_1,\dots, x_n)\in \mathcal {A}<^>n $ x=(x1,& mldr;,xn)is an element of An. Furthermore, we complement and explain more about the case when n = 1.

Let L(3 ) be the algebra of all bounded linear operators acting on a separable infinite-dimensional complex Hilbert space 3 , and denote by w(T) the numerical radius of any operator T is an element of L(3 ). In this paper, we describe the form of all bijective linear maps phi on L(3 ) for which w(phi(T)) = w(phi(S)) whenever T, S is an element of L(3 ) are two unitarily similar operators. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let $ \mathcal {B}({\mathcal H}) $ B(H) be the algebra of all bounded linear operators acting on a complex Hilbert space $ {\mathcal H} $ H. For an operator $ T\in \mathcal {B}({\mathcal H}) $ T is an element of B(H) with a polar decomposition $ T= \mathcal {V}_T \vert T\vert $ T=VT|T|, its mean transform is defined by \[ \mathcal{M}(T):=\frac{1}{2}\left(\mathcal{V}_T|T|+|T|\mathcal{V}_T\right). \]M(T):=12(VT|T|+|T|VT). In this paper, we obtain the form of all bijective linear maps Phi on $ \mathcal {B}({\mathcal H}) $ B(H) for which $ \mathcal {M}(\Phi (T)) $ M(Phi(T)) and $ \mathcal {M}(\Phi (S)) $ M(Phi(S)) are unitarily similar whenever $ T, S\in \mathcal {B}({\mathcal H}) $ T,S is an element of B(H) are two unitarily similar operators. To achieve this, we first characterize all surjective maps Phi on $ \mathcal {B}({\mathcal H}) $ B(H) satisfying (1) $$\begin{equation} \mathcal{M}\left(\Phi(T)-\Phi(S)\right)\sim_u \mathcal{M}(T-S),\quad (T, S\in\mathcal{B}({\mathcal H})). \end{equation}$$M(Phi(T)-Phi(S))similar to uM(T-S),(T,S is an element of B(H)). Furthermore, a number of related results and consequences is obtained.

Let ℒ(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ . For an operator T∈ℒ(ℋ) , let W_0(T) be the maximal numerical range of T. We show that a map φ from ℒ(ℋ) onto itself satisfies W_0( φ (S)φ (T)φ (S)) = W_0(STS), (T, S∈ℒ(ℋ)), if and only if there are a unitary operator U∈ℒ(ℋ) and λ∈ℂ such that λ ^3=1 and either φ (T)= λ UTU^* for all T∈ℒ(ℋ) , or φ (T)= λ UT^⊤ U^* for all T∈ℒ(ℋ) . Here, T^⊤ denotes the transpose of any operator T∈ℒ(ℋ) relative to a fixed but arbitrary orthonormal base of ℋ . When the triple product “STS” is replaced by the skew-triple product “ TS^*T ”, we arrive at the same conclusion but with λ =1 .

Let B ( H ) be the algebra of all bounded linear operators acting on an infinite- dimensional separable complex Hilbert space H . The polar decomposition theorem asserts that every operator T E B ( H ) can be uniquely written as T = V T | T L the product of a partial isometry V T E B ( H ) that has the same kernel as that of T and the modulus |T (T*T)12of T . Given a scalar ) E [0, 1], the )-Aluthge transform of any T E B ( H ) is Da(T) :=|T|aVT|T|1-a. In this paper, we obtain the form of all bijective linear maps I' on B ( H ) for which Da (I'(T)) and Da (I'(S)) are unitarily similar whenever T, S E B ( H )are unitarily similar. To achieve this, we characterize all maps I' on B ( H ) for which Da (phi(T)- I'(S)) and Da (T- S ) are unitarily similar for all T, S E B ( H ). Moreover, we obtain the form of all bijective linear maps I' on B ( H ) for which I' (Da(T)) and Da (I'(S)) are unitarily similar whenever T, S E B ( H ) are unitarily similar. Furthermore, a number of related results and consequences is obtained. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let L(H) be the algebra of all bounded linear operators on a complex Hilbert space H. For any operator T is an element of L (H), let W-0(T) be the maximal numerical range of T. We show that two maps phi(1) and phi(2) from L (H) onto itself satisfy W-0 (phi(1)(S)phi(2)(T)) = W-0(ST ), (T, S is an element of L ( H )) , if and only if there are two invertible operators A, U is an element of L ( H ) such that U is unitary and phi(1)( T ) = UTA for all T is an element of L (H) and phi(2) (T) = A(-1)TU(& lowast;) for all T is an element of L (H). Some known results are obtained as immediate consequences.

Let B ( H ) \mathcal {B}(H) be the algebra of all bounded linear operators on a complex Hilbert space H H . For any unit vector x ∈ H x\in H , let P x P_x be the orthogonal projection onto the span of x x . For any T ∈ B ( H ) T\in \mathcal {B}(H) , let γ ( T , x ) ≔ sup { t ≥ 0 : t P x ≤ | T | } \gamma (T,x)≔\sup \left \{ t\geq 0 \; : \; t P_x \leq \vert T\vert \right \} be the local reduced minimum modulus of T T at x x and set Λ ( T ) ≔ { γ ( T , x ) : x ∈ H , ‖ x ‖ = 1 } \Lambda (T)≔\left \{ \gamma (T, x): \quad x\in H,\;\|x\|=1\right \} . In this paper, we investigate surjective linear maps on B ( H ) \mathcal {B}(H) compressing this set.

Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. For any operator A is an element of B(H), let |A|:=(A & lowast;A)( 1 /2) and A dagger be the modulus and the Moore-Penrose inverse of A, respectively. The polar factor of an operator A is an element of B(H) is, by the polar decomposition theorem, the unique partial isometry VA is an element of B(H) whose kernel coincides with that of A such that A = VA|A|. In this paper, we show that if A is an element of B(H) has closed range then the sequence 1 X-0 = A, Xn+1 =1/2 (X-n + X-n & lowast;dagger ) , (n > 0), converges to V-A in the norm topology of B(H). This extends Higham's matrix result to the setting of operators with closed range. Furthermore, we obtain another approximation of the polar factors of operators in B(H)and discuss a related linear preserving problem. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

be the M-reduced minimum modulus of T . In this paper, we first establish certain permanence properties of this quantity, and express it in term of the Moore-Penrose inverse T dagger of T . Then we discuss the relationship between the M-reduced minimum moduli of any T E B ( H ). We also investigate the relationship between the Mreduced minimum modulus of any operator T E B ( H ) and the minimum modulus of its restriction to M . For a fixed operator T 0 E B ( H ), the M-reduced minimum moduli of T 0 is a function on the set of all closed subspaces M of H . When such a set is equipped with the gap distance, we show, in particular, that this function is upper semi-continuous, and then prove that it is continuous if and only if T 0 is a left invertible operator. We also consider, for a fixed closed linear subspace M 0 of H , the function T -> 7(T, M 0 ) and discuss its points of continuity. As applications, we characterize, in terms of the M-reduced minimum moduli, several classes of operators such as semi-Fredholm and posinormal operators. Finally, we close this paper by describing the M-reduced minimum moduli of weighted shift operators and their adjoints. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let n be an integer greater than 1, and Mn(C) be the algebra of all n×n-complex matrices. Let x0∈Cn be a nonzero vector, and Φ be a linear map on Mn(C) such that Φ(I) is invertible. For any matrix T∈Mn(C), let γ(T,x0) denote the local reduced minimum modulus of T at x0. In this paper, we show that Φ satisfiesγ(T,x0)=0⇔γ(Φ(T),x0)=0,(T∈Mn(C)), if and only if there are two invertible matrices A,B∈Mn(C) such that Ax0=A⁎x0=x0 and Φ(T)=BTA for all T∈Mn(C). When n=2, we show that the invertibility hypothesis of Φ(I) is redundant.

Let ℒ(ℋ) be the algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space ℋ , and denote by γ (T) the reduced minimum modulus of any operator T∈ℒ(ℋ) . We obtain the form of all bijective linear maps Φ on ℒ(ℋ) for which γ (Φ (T))=γ (Φ (S)) whenever T, S∈ℒ(ℋ) are two operators equivalent by unitaries. We also obtain similar results when the reduced minimum modulus is replaced by the minimum modulus or the surjectivity modulus.

Let ℳ_n(ℂ) be the algebra of all n× n complex matrices, and denote by V ( T ) and v ( T ) the numerical range and numerical radius of any matrix T∈ℳ_n(ℂ) , respectively. Let ϕ : ℳ_m(ℂ)⟶ℳ_n(ℂ) be a nonsurjective linear maps that satisfy v(ϕ (A))=v(A) for all A∈ℳ_m(ℂ) . In this note, we show that if m≥ 2 then ϕ (V) is spectraloid for any unitary matrix V∈ℳ_m(ℂ) . Also, under the mild condition that ‖ϕ (1_m)‖≤ 1 , we prove that then there are unimodular scalars λ _1, … , λ _p in the spectrum of ϕ (1_m) such that V(A)⊂ V( ⊕ _k=1^pλ_kϕ (A)) for all A∈ℳ_m(ℂ) .

Let 13(H) be the algebra of all bounded linear operators on a complex Hilbert space H. For any operator T & ISIN; 13(H) and unit vector x & ISIN; H, let-y(T, x) denote the local reduced minimum modulus of T at x. In this paper, we characterize surjective maps on 13(H) such that, for all S, T & ISIN; 13(H) and all unit vectors x & ISIN; H, one has-y(T - S, x) = 0 if and only if-y(& phi;(T) - & phi;(S), x) = 0. We also describe the form of all maps on 13(H) preserving the product of operators of zero local reduced minimum moduli. Furthermore, some consequences and open problems are also discussed. & COPY; 2023 Elsevier Inc. All rights reserved.

Let $$\mathfrak {A}$$ be a unital $$C^*$$ -algebra and $$\mathfrak {A}'$$ be its topological dual space. Let a be a positive element in $$\mathfrak {A}$$ , and set $$\mathscr {S}_a(\mathfrak {A}):= \left\{ f\in \mathfrak {A}': f\ge 0, f(a)=1\right\} .$$ The a-numerical range and a-numerical radius of any element $$x\in \mathfrak {A}$$ are defined by $$\begin{aligned} V_a(x):= \left\{ f(ax): f\in \mathscr {S}_a(\mathfrak {A})\right\} , \end{aligned}$$ and $$\begin{aligned} v_a(x):=\sup \left\{ \left| z\right| :z\in V_a(x)\right\} , \end{aligned}$$ respectively. In this paper, we establish some permanence properties of the a-numerical range and a-numerical radius of elements in $$\mathfrak {A}$$ . In particular, we investigate when the a-numerical range of an element of $$\mathfrak {A}$$ is closed, and provide explicit formulas for the a-numerical radius of the so-called a-hermitian elements of $$\mathfrak {A}$$ . Furthermore, given a positive operator A on a complex Hilbert space $${\mathscr {H}}$$ , we study and investigate the relationship between the algebraic and spatial A-numerical ranges of bounded linear operators on $${\mathscr {H}}$$ .