Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form [{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}], where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that is, for some fixed $k$, $1\le k \le \lfloor n/2\rfloor$, $n$ is divisible by $k$, and $|a_j|=|a_{k+j}|$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1...a_n=b_1...b_n$ and $S_r(|a_1|^2,..., |a_n|^2)=S_r(|b_1|^2,..., |b_n|^2)$ for all $1\le r\le \lfloor n/2\rfloor$, where $S_r$'s are the circularly symmetric functions.Comment: 27 page

Intra-medullary abscess of the spinal cord at the high cervical level

SummaryIntramedullary spinal cord abscess (ISCA) is a rare entity in the central nervous system, especially in patients without meningitis. It is often difficult to promptly establish the diagnosis. In this report, we describe the case of a woman 52 years of age who presented with progressive neck pain that was followed by rapidly worsening quadriparesis. The cervical spine magnetic resonance imaging revealed an ovoid lesion at the level of C2-C4 with a gadolinium contrast ring enhancement. Because ISCA was suspected, we performed surgery to decompress the cervical spinal cord. The bacterial culture of the cystic content revealed the growth of two bacteria, Fusobacterium nucleatum and Peptoniphilus asaccharolyticus, which have rarely been observed. The patient's postoperative course was uneventful and her outcome was excellent on discharge. Antibiotics were prescribed for 6 weeks. We further report the result of a literature review on the management of ISCA. The surgical indications for ISCA are controversial, but early surgery to decompress the spinal cord and confirm bacterial growth is recommended in rapidly worsening cases

Factoring a Quadratic Operator as a Product of Two Positive Contractions

Let T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the form aI circle plus bI circle plus [ GRAPHICS ] on H-1 circle plus H-2 circle plus (H-3 circle plus H-3) for some a, b is an element of [ 0, 1 ] and strictly positive operator P with parallel to P parallel to \u3c = vertical bar root a - root b vertical bar root (1 - a) (1 - b). Also, we give a necessary condition for a bounded linear operator T with operator matrix [GRAPHICS] on H circle plus K that can be written as a product of two positive contractions