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We study the maximal algebraic degree of principal ortho-compressions of linear operators that constitute spatial matricial numerical ranges of higher order. We demonstrate (amongst other things) that for a (possibly unbounded) operator L L on a Hilbert space, every principal m m -dimensional ortho-compression of L L has algebraic degree less than m m if and only if r a n k ( L − λ I ) ≤ m − 2 rank(L-\lambda I)\le m-2 for some λ ∈ C \lambda \in \mathbb {C} .

We explore the relationship between local and global reducibility of spaces of nilpotent matrices. By local reducibility we mean that small subspaces of a given irreducible linear space L ⊆ M n ( C ) are reducible. One of our main results is that for certain integers m depending on n there is an ( m + 1 ) -dimensional space L which is irreducible, but every one of its m-dimensional subspaces is, not just reducible, but simultaneously triangularizable.

Let $\mathcal{H}$ be a complex, separable Hilbert space, and $\mathcal{B}(\mathcal{H})$ denote the set of all bounded linear operators on $\mathcal{H}$. Given an orthogonal projection $P \in \mathcal{B}(\mathcal{H})$ and an operator $D \in \mathcal{B}(\mathcal{H})$, we may write $D=\begin{bmatrix} D_1& D_2 D_3 & D_4 \end{bmatrix}$ relative to the decomposition $\mathcal{H} = \mathrm{ran}\, P \oplus \mathrm{ran}\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\mathrm{rank}\, D_2 = j$ and $\mathrm{rank}\, D_3 = k$? Complete results are obtained in the case where $\mathrm{dim}\, \mathcal{H} < \infty$, and partial results are obtained in the infinite-dimensional setting.

Let n ∈ N , and consider C n equipped with the standard inner product. Let A ⊆ L ( C n ) be a unital algebra and P ∈ L ( C n ) be an orthogonal projection. The space L : = P ⊥ A | ran P is said to be an off-diagonal corner of A , and L is said to be essential if ∩ { ker ⁡ L : L ∈ L } = { 0 } and ∩ { ker ⁡ L ⁎ : L ∈ L } = { 0 } , where L ⁎ denotes the adjoint of L. Our goal in this paper is to determine effective upper bounds on dim ⁡ A in terms of dim ⁡ L , where L is an essential off-diagonal corner of A . A detailed structure analysis of A based upon the dimension of L , while seemingly elusive in general, is nevertheless provided in the cases where dim ⁡ L ∈ { 1 , 2 } .

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group.

We give a short proof of a theorem of Mirsky on doubly substochastic matrices and prove generalizations.

We call a function of k variables reducing for a multiplicative semigroup S ⊆ M n ( C ) , if the semigroup is reducible whenever f sends S to zero. For continuous reducing functions on groups of unitary matrices we prove that such a group is reducible whenever f is uniformly small enough on the group. Extensions of this result for certain types of semigroups are also given. Note that our results contain and extend many known results of the kind. In particular, it is shown that a group of complex unitary matrices is reducible if, for fixed indices i and j, its ( i , j ) entries are uniformly small.

Let M n be the algebra of all n × n complex matrices, 1 ≤ k ≤ n − 1 be an integer, and φ : M n ⟶ M n be a linear operator. In this paper, it is shown that φ preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix U ∈ M n such that either φ ( A ) = U ⁎ A U for all A ∈ M n , or φ ( A ) = U ⁎ A t U for all A ∈ M n .

We extend a well-known theorem of Burnside in the setting of general fields as follows: for a general field F the matrix algebra $$M_n(F)$$ is the only algebra in $$M_n(F)$$ which is spanned by an irreducible semigroup of triangularizable matrices. In other words, for a semigroup of triangularizable matrices with entries from a general field irreducibility is equivalent to absolute irreducibility. As a consequence of our result we prove a stronger version of a theorem of Janez Bernik.

Let \(\mathcal S\) be a multiplicative semigroup of bounded linear operators on a complex Hilbert space \(\mathcal H\), and let \(\Omega \) be the range of a vector state on \(\mathcal S\) so that \(\Omega = \{ \langle S \xi , \xi \rangle \,{:}\,S \in \mathcal S\}\) for some fixed unit vector \(\xi \in \mathcal H\). We study the structure of sets \(\Omega \) of cardinality two coming from irreducible semigroups \(\mathcal S\). This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for \(\mathcal S\). This is made possible by a thorough investigation of the structure of maximal families \(\mathcal F\) of unit vectors in \(\mathcal H\) with the property that there exists a fixed constant \(\rho \in \mathbb C\) for which \(\langle x, y \rangle = \rho \) for all distinct pairs x and y in \(\mathcal F\).

Let S be a semigroup of invertible matrices. It is shown that if P is an idempotent matrix of rank and co-rank at least two such that the rank of ( 1 − P ) S P is never more than one for S in S (the range of the kind of P is said to be near-invariant), then S has an invariant subspace within one dimension of the range of P (the kind of range is said to be nearly invariant).

One consequence of the Perron–Frobenius Theorem on indecomposable positive matrices is that whenever an $$n\times n$$ matrix A dominates a non-singular positive matrix, there is an integer k dividing n such that, after a permutation of basis, A is block-monomial with $$k\times k$$ blocks. Furthermore, for suitably large exponents, the nonzero blocks of $$A^m$$ are strictly positive. We present an extension of this result for indecomposable semigroups of positive matrices.

An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be \textit{idempotent compressible} if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Analogously, $\mathcal{A}$ is said to be \textit{projection compressible} if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P$ in $\mathbb{M}_n(\mathbb{C})$. In this paper we construct several examples of unital algebras that admit these properties. In addition, a complete classification of the unital idempotent compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ is obtained up to similarity and transposition. It is shown that in this setting, the two notions of compressibility agree: a unital subalgebra of $\mathbb{M}_3(\mathbb{C})$ is projection compressible if and only if it is idempotent compressible. Our findings are extended to algebras of arbitrary size in the sequel to this paper., Comment: 24 pages

A subalgebra $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Likewise, $\mathcal{A}$ is said to be idempotent compressible if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. In this paper, a case-by-case analysis is used to classify the unital projection compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$, up to transposition and unitary equivalence. It is observed that every algebra shown to admit the projection compression property is, in fact, idempotent compressible. We therefore extend the findings of Cramer, Marcoux, and Radjavi (arXiv:1904.06803 [math.RA]) in the setting of $\mathbb{M}_3(\mathbb{C})$, proving that the two notions of compressibility agree for all unital matrix algebras., Comment: 36 pages

First published in Transactions of the American Mathematical Society in volume 368, 2016, published by the American Mathematical Society. https://doi.org/10.1090/tran/6619

Let $$\varphi $$ be a linear functional of rank one acting on an irreducible semigroup $$\mathcal {S}$$ of operators on a finite- or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of $$\varphi $$ cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional $$\varphi $$ with a two-element range yields valuable information on the structure of $$\mathcal {S}$$ .

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also obtain a complete characterization of those operators for which $\mathrm{rank}\, (I-P) T P = \mathrm{rank}\, P T (I-P)$ for all orthogonal projections $P$. When $\mathcal{H}$ is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane., 24 pages

In this article we verify that ‘Wedderburn’s Principal Theorem’ has a particularly pleasant spatial implementation in the case of cleft subalgebras of the algebra of all linear transformations on a finite-dimensional vector space. Once such a subalgebra is represented by block upper triangular matrices with respect to a maximal chain of its invariant subspaces, after an application of a block upper triangular similarity, the resulting algebra is a linear direct sum of an algebra of block-diagonal matrices and an algebra of strictly block upper triangular matrices (i.e. the radical), while the block-diagonal matrices involved have a very nice structure. We apply this result to demonstrate that, when the underlying field is algebraically closed, and , the algebra is unicellular, i.e. the lattice of all invariant subspaces of is totally ordered by inclusion. The quantity stands for the length of (every) maximal chain of non-zero invariant subspaces of .

We consider matrix semigroups S which are closed under multiplication by complex scalars, and whose norm closure contains no zero-divisors. We show that when every non-zero S in S is indecomposable and the spectral radius of S is equal to the spectral radius of |S| for all S in S, then S is effectively positive, in the sense that there exists a diagonal unitary matrix D so that for each S in S, S = αS D|S|D −1 for some αS ∈ T. We also show the same conclusion holds even if individual indecomposability is weakened to indecomposability of the semigroup as a whole, as long as the semigroup is convex. We give examples showing that all hypotheses are required. We also extend some of these results to compact operators, under additional conditions.

In this article we study a natural weakening – which we refer to as paratransitivity – of the well-known notion of transitivity of an algebra A of linear operators acting on a finite-dimensional vector space V. Given positive integers k and m, we shall say that such an algebra A is (k,m)-transitive if for every pair of subspaces W1 and W2 of V of dimensions k and m respectively, we have AW1∩W2≠{0}. We consider the structure of minimal (k,m)-transitive algebras and explore the connection of this notion to a measure of largeness for invariant subspaces of A.

A closed subspace Y of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T ∈ S there exists a finite-dimensional subspace F T of X such that T Y ⊆ Y + F T . In this paper, we study the existence of almost-invariant subspaces for algebras of operators. We show, in particular, that if a closed algebra of operators on a Hilbert space has a non-trivial almost-invariant subspace then it has a non-trivial invariant subspace. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if T is an operator on a separable Hilbert space and if T P − P T has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T = M + F where M ∈ M and F is of finite rank.

We present several extensions of Burnside’s well-known theorem which states that the only irreducible subalgebra of M n ( F ) with algebraically closed field F is M n ( F ) itself. We show, among some stronger results, that if F is quasi-algebraically closed (in particular, if F is finite), then the only irreducible subalgebra of M n ( F ) that contains a linear basis of triangularizable matrices (a hypothesis that automatically holds in the classical case) is M n ( F ) itself. We also consider the problem of “field of definition” for a semigroup S in M n ( K ) : If a linear functional on S takes values in a smaller field F, is S simultaneously similar to a semigroup in M n ( F ) ?

Let P be a projection and let S be a multiplicative semigroup of linear operators such that SP - PS is nilpotent for every S in S . We study conditions under which this implies the existence of an invariant subspace for S and, in particular, when P commutes with every member of S .

A set [Formula: see text] of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists [Formula: see text] such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

Let S be a multiplicative semigroup of matrices with nonnegative entries. Assume that the diagonal entries of the members of S form a finite set. This paper is concerned with the following question: Under what circumstances can we deduce that S itself is finite?

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in $${\mathcal{S}}$$ satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.

We establish finite- and infinite-dimensional versions of the following assertion. If M is a matrix with the property that whenever P and Q are diagonal projections with P⩽Q, the spectrum of PMP (considered as an operator on the range of P) is contained in that of QMQ (considered as an operator on the range of Q), then there is a permutation matrix U such that U−1MU is triangular.

A semigroup S of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists S ∈ S such that (S) ij �= 0. We show that if there is a pair k, l such that {(S)kl : S ∈ S} is bounded then, after a simultaneous diagonal similarity, all the entries are in (0,1). We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable., 14 pages

Motivated by problems concerning simultaneous triangularization, we study the structure of finite, minimal nonabelian groups. Using the structure result of Miller and Moreno we explicitly describe all irreducible representations of such groups. We illustrate the usefulness of results of this type on several examples.

Let f be a noncommutative polynomial in two variables. Let be a multiplicative semigroup of linear operators on a finite-dimensional vector space and T a fixed linear operator such that f(T, S) is nilpotent for all S in . What can we say about the invariant subspace structure of We study special cases of this and other related conditions.

The operators on L p ,1 p

We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson's result on the weak-operator density of transitive spaces that are masa bimodules., 23 pages

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x n ) in B there exists a subsequence \((x_{{n}_{k}})\) and a bounded sequence (A k ) in the algebra such that \((A_{k}x_{{n}_{k}})\) converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every \(\varepsilon > 0\) there exists A in the algebra such that \(||Ax - y|| < \varepsilon\) and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings.

Let X and Y be Banach spaces and B(X) and B(Y ) the algebras of all bounded linear operators on X and Y , respectively. We say that A,B 2 B(X) quasi- commute if there exists a nonzero scalar ! such that AB = !BA. We characterize bijective linear maps � : B(X) ! B(Y ) preserving quasi-commutativity. In fact, such a character- ization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

We prove a semigroup analogue of the Kadison Transitivity Theorem for C ∗ -algebras. Specifically, we show that a closed, homogeneous, self-adjoint, topologically transitive, semigroup of operators acting on a separable Hilbert space is (strictly) transitive if the semigroup contains a non-zero compact operator. Additional structural information about such semigroups is obtained, and examples are provided to demonstrate that the theorem is the best possible in the semigroup case.

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky's Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings. Also, we give a generalization of Kaplansky's Theorem over general fields. We show that this extension of Kaplansky's Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky's Theorem holds over $\Delta$., Comment: arXiv admin note: text overlap with arXiv:1508.00183

If a nonzero linear functional has finite, countable, or bounded range when restricted to an irreducible semigroup \({\mathcal{S}}\) of complex matrices, it is shown that \({\mathcal{S}}\) itself has the same property. Similar results are proven under the hypothesis that a nontrivial ideal of \({\mathcal{S}}\) is finite, countable, or bounded.