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Register a new userThe Gau-Wang-Wu conjecture on partial isometries holds in the 5-by-5 case
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Gau, Wang and Wu in their LAMA2016 paper conjectured (and proved for $nleq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.
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In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $ntimes n$ matrix $A$, which counts the maximal number of orthonormal vectors $textbf x_j$ such that the scalar products $langle Atextbf x_j,textbf x_jrangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu numbers for $3times 3$ matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified $W(A)$ for $4times 4$ unitarily irreducible $A$ with irreducible base curve according to singularities of that curve. When $A$ is a unitarily irreducible $ntimes n$ matrix, we give necessary conditions for $k(A) = 2$, characterize $k(A) = n$, and apply these results to the case of unitarily irreducible $4times 4$ matrices. However, we show that knowledge of the singularities is not sufficient to determine $k(A)$ by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different $k(A)$. In addition, we extend Chien and Nakazatos classification to consider unitarily irreducible $A$ with reducible base curve and show that we can find corresponding matrices with identical base curve but different $k(A)$. Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky, generalizing their result in the process.
The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau--Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalar products $langle Ax_j,x_jrangle$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4times4$ matrices according to the values of their Gau--Wu numbers.
We show that ||u*u - v*v|| leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.
A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $Delta_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that $Delta_{sm}(4,9)=5$ and $Delta_m(4,9)=Delta_m(5,10)=6.$ We shorten the known non-computer proof of the strict monotone 4-step conjecture.
In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kahler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.
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