Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write \(\mathcal P\mathcal T_{XY}\) for the set of all partial functions \(X\rightarrow Y\). Fix a partial function \({a:Y\rightarrow X}\), and define the operation \(\star _a\) on \(\mathcal P\mathcal T_{XY}\) by \(f\star _ag=fag\) for \(f,g\in \mathcal P\mathcal T_{XY}\). The sandwich semigroup \((\mathcal P\mathcal T_{XY},\star _a)\) is denoted \(\mathcal P\mathcal T_{XY}^a\). We apply general results from Part I to thoroughly describe the structural and combinatorial properties of \(\mathcal P\mathcal T_{XY}^a\), as well as its regular and idempotent-generated subsemigroups, \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). After describing regularity, stability and Green’s relations and preorders, we exhibit \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups \(\mathcal P\mathcal T_X\) and \(\mathcal P\mathcal T_Y\), and as a kind of “inflation” of \(\mathcal P\mathcal T_A\), where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of \(\mathcal P\mathcal T_{XY}^a\), \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Counting exceptional points for rational numbers associated to the Fibonacci sequence

If \(\alpha \) is a non-zero algebraic number, we let \(m(\alpha )\) denote the Mahler measure of the minimal polynomial of \(\alpha \) over \(\mathbb Z\). A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the t-metric Mahler measure, denoted \(m_t(\alpha )\). For fixed \(\alpha \in \overline{\mathbb Q}\), the map \(t\mapsto m_t(\alpha )\) is continuous, and moreover, is infinitely differentiable at all but finitely many points, called exceptional points for \(\alpha \). It remains open to determine whether there is a sequence of elements \(\alpha _n\in \overline{\mathbb Q}\) such that the number of exceptional points for \(\alpha _n\) tends to \(\infty \) as \(n\rightarrow \infty \). We utilize a connection with the Fibonacci sequence to formulate a conjecture on the t-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

On some spectral spaces associated to tensor triangulated categories

We consider a closure operator c of finite type on the space \(SMod(\mathcal M)\) of thick \(\mathcal K\)-submodules of a triangulated category \(\mathcal M\) that is a module over a tensor triangulated category \((\mathcal K,\otimes ,1)\). Our purpose is to show that the space \(SMod^c(\mathcal M)\) of fixed points of the operator c is a spectral space that also carries the structure of a topological monoid.     

Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation \({{\mathrm{div}}}(\sigma \nabla u) = 0\) in dimension two, in the case when the essential range of \(\sigma \) consists of only two elliptic matrices, i.e., \(\sigma \in \{\sigma _1, \sigma _2\}\) a.e. in \(\Omega \). In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices \(\sigma _1\) and \(\sigma _2\), exponents \(p_{\sigma _1,\sigma _2}\in (2,+\infty )\) and \(q_{\sigma _1,\sigma _2}\in (1,2)\) have been found so that if \(u\in W^{1,q_{\sigma _1,\sigma _2}}(\Omega )\) is solution to the elliptic equation then \(\nabla u\in L^{p_{\sigma _1,\sigma _2}}_{\mathrm{weak}}(\Omega )\) and the optimality of the upper exponent \(p_{\sigma _1,\sigma _2}\) has been proved. In this paper we complement the above result by proving the optimality of the lower exponent \(q_{\sigma _1,\sigma _2}\). Precisely, we show that for every arbitrarily small \(\delta \), one can find a particular microgeometry, i.e., an arrangement of the sets \(\sigma ^{-1}(\sigma _1)\) and \(\sigma ^{-1}(\sigma _2)\), for which there exists a solution u to the corresponding elliptic equation such that \(\nabla u \in L^{q_{\sigma _1,\sigma _2}-\delta }\), but \(\nabla u \notin L^{q_{\sigma _1,\sigma _2}}\). The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.     

Congruences for $$\ell $$-regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.     

Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball \(\Omega _n\) is the set of all \(n\times n\) matrices M with spectral radius less than 1. Let \(\pi (M) \in \mathbb {C}^n\) stand for the coefficients of the characteristic polynomial of a matrix M (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisc is \({{\mathbb {G}}}_n:=\pi (\Omega _n)\). When investigating Nevanlinna–Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisc (for instance to obtain continuity results for the Lempert function): if \(\Phi \in {\mathrm {Hol}}(\mathbb {D}, \Omega _n)\), then \(\pi \circ \Phi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\). Given a map \(\varphi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\), we are looking for necessary and sufficient conditions for this map to “lift through given matrices”, i.e. find \(\Phi \) as above so that \(\pi \circ \Phi = \varphi \) and \(\Phi (\alpha _j) = A_j\), \(1\le j \le N\). A natural necessary condition is \(\varphi (\alpha _j)=\pi (A_j)\), \(1\le j \le N\). When the matrices \(A_j\) are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of \(\varphi \) at the points \(\alpha _j\). We prove that those conditions are necessary and sufficient for a local lifting. We give a formula which performs the global lifting in small dimensions (\(n \le 5\)), and a counter-example to show that the formula fails in dimensions 6 and above.     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

Congruences modulo 16, 32, and 64 for Andrews’s singular overpartitions

In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function \(\overline{C}_{k,i}(n) \) which denotes the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i \ (\mathrm{mod}\ k)\) may be overlined. Andrews, Chen, Hirschhorn and Sellers, and Ahmed and Baruah discovered numerous congruences modulo 2, 3, 4, 8, and 9 for \(\overline{C}_{3,1}(n)\). In this paper, we prove a number of congruences modulo 16, 32, and 64 for \(\overline{C}_{3,1}(n)\).     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

The Difference Between a Discrete and Continuous Harmonic Measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let \(\omega _h(0,\cdot ;D)\) be the discrete harmonic measure at \(0\in D\) associated with this random walk, and \(\omega (0,\cdot ;D)\) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function \(\sigma _D(z)\) on \(\partial D\) such that for functions g which are in \(C^{2+\alpha }(\partial D)\) for some \(\alpha >0\) we have

$$\begin{aligned} \lim _{h\downarrow 0} \frac{\int _{\partial D} g(\xi ) \omega _h(0,|\mathrm{d}\xi |;D) -\int _{\partial D} g(\xi )\omega (0,|\mathrm{d}\xi |;D)}{h} = \int _{\partial D}g(z) \sigma _D(z) |\mathrm{d}z|. \end{aligned}$$

We give an explicit formula for \(\sigma _D\) in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

     

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to \({\mathbb {R}}^n\). The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by \(C(K,{\mathbb {R}}^n)\) the set of continuous maps from K to \({\mathbb {R}}^n\) endowed with the maximum norm. Let \(\dim _{*}\) be one of the topological dimension \(\dim _T\), the Hausdorff dimension \(\dim _H\), or the packing dimension \(\dim _P\). Define

$$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$

We prove that \(d^n_{*}(K)\) is the right notion to describe the dimensions of the fibers of a generic continuous map \(f\in C(K,{\mathbb {R}}^n)\). In particular, we show that \(\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)\) provided that \(\dim _T K\ge n\), otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume \(\dim _T K\ge n\). If K is sufficiently homogeneous, then we can say much more. For example, we prove that \(\dim _{*}f^{-1}(y)=d^n_{*}(K)\) for a generic \(f\in C(K,{\mathbb {R}}^n)\) for all \(y\in {{\mathrm{int}}}f(K)\) if and only if \(d^n_{*}(U)=d^n_{*}(K)\) or \(\dim _T U<n\) for all open sets \(U\subset K\). This is new even if \(n=1\) and \(\dim _{*}=\dim _H\). It is known that for a generic \(f\in C(K,{\mathbb {R}}^n)\) the interior of f(K) is not empty. We augment the above characterization by showing that \(\dim _T \partial f(K)=\dim _H \partial f(K)=n-1\) for a generic \(f\in C(K,{\mathbb {R}}^n)\). In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.

     

Overpartition function modulo powers of 2

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\),

$$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

     

Nordhaus–Gaddum Results for the Induced Path Number of a Graph When Neither the Graph Nor Its Complement Contains Isolates

The induced path number \(\rho (G)\) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. A product Nordhaus–Gaddum-type result is a bound on the product of a parameter of a graph and its complement. Hattingh et al. (Util Math 94:275–285, 2014) showed that if G is a graph of order n, then \(\lceil \frac{n}{4} \rceil \le \rho (G) \rho (\overline{G}) \le n \lceil \frac{n}{2} \rceil \), where these bounds are best possible. It was also noted that the upper bound is achieved when either G or \(\overline{G}\) is a graph consisting of n isolated vertices. In this paper, we determine best possible upper and lower bounds for \(\rho (G) \rho (\overline{G})\) when either both G and \(\overline{G}\) are connected or neither G nor \(\overline{G}\) has isolated vertices.     

Perfect state transfer in Laplacian quantum walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix \(U(t) = \exp (-itM)\), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time \(\tau \) if the (u, v)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if \(m \equiv 2\pmod {4}\). This was previously known for the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph G has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of G and \(\mu \) of H, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of \(P_{3}\) has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129–147, 2011).     

On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).