User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:18:25Z http://mathoverflow.net/feeds/user/13481 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120980/possible-restrictions-on-generators-of-m-n-mathbbc Possible restrictions on generators of $M_n(\mathbb{C})$ spelas 2013-02-06T15:55:10Z 2013-02-06T18:29:54Z <p>Suppose matrices $a$ and $b$ generate $M_n(\mathbb{C})$. I would like to know what restrictions this imposes on $a$ and $b$. More concretely, do there exist $a,b\in M_n(\mathbb{C})$, which generate $M_n(\mathbb{C})$, such that the minimal polynomial of $\alpha a+\beta b$ has degree $\leq n-1$ for all $\alpha,\beta\in \mathbb{C}$?</p> http://mathoverflow.net/questions/116821/centralizer-in-a-matrix-algebra-over-commutative-polynomials Centralizer in a matrix algebra over commutative polynomials spelas 2012-12-19T21:46:04Z 2012-12-19T21:46:04Z <p>Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$. I would like to know what is the centralizer of the matrix $x=(x_{ij})$ in $A$.</p> <p>If we consider $x$ as an element of $M_n(F(x_{ij}\mid 1\leq i,j\leq n))$, it is not difficult to show that its centralizer consists of polynomials in $x$ with coefficients in $F(x_{ij}\mid 1\leq i,j\leq n)$. Does the centralizer of $x$ in $A$ consists of polynomials in $x$ with coefficients in $F[x_{ij}\mid 1\leq i,j\leq n]$?</p> http://mathoverflow.net/questions/84863/centralizers-in-c-algebra Centralizers in C*-algebra spelas 2012-01-04T06:20:39Z 2012-02-05T05:46:28Z <p>Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, <code>$\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$</code>. Can we say anything about the correspondence between $a$ and $b$?</p> <p>For example, if $A=B(H)$ for a separable Hilbert space $H$, then according to the double commutant theorem $b=f(a)$ for some Borel function $f$ on the spectrum of $a$.</p> http://mathoverflow.net/questions/64107/spectra-of-sums-in-banach-algebras spectra of sums in (Banach) algebras spelas 2011-05-06T10:30:36Z 2012-01-04T06:38:37Z <p>A similar question was already asked in <a href="http://mathoverflow.net/questions/14246/spectra-of-sums-and-products-in-banach-algebras-was-spectrum-in-banach-algebr" rel="nofollow">question</a> titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".</p> <p>Answer there led me to the following question.</p> <p>If for elements $a,b$ in Banach algebra $A$ hold that $\operatorname{spec}(\lambda a+\mu b)\subseteq \lambda \operatorname{spec}(a)+\mu \operatorname{spec}(b)$ for every $\lambda, \mu\in \mathbb{C}$, can we say something about the subalgebra $B$ of $A$ generated by $a$ and $b$. Might we conclude something about $B/\operatorname{rad}(B)$?</p> <p>Where to look for (possible) counterexample of $B/\operatorname{rad}(B)$ being commutative?</p> <p>Thank you for finding a counterexample. Now I would like to add some additional assumptions. Suppose that for a selfadjoint element $a$ and a unitary element $u$ of $C^∗$-algebra holds that $\operatorname{spec}(a+\lambda a^2)=\operatorname{spec}(a+\lambda ua^2u^∗)$ for every $λ∈\mathbb{C}$. Moreover, if $\lambda \in \mathbb{R}$ elements $a+\lambda a^2$ and $a+\lambda ua^2u^∗$ are unitarily equivalent.</p> http://mathoverflow.net/questions/79461/positive-hermitian-elements-in-m-n-mathbbc positive hermitian elements in $M_n(\mathbb{C})$ spelas 2011-10-29T12:01:22Z 2011-10-30T10:40:36Z <p>Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers have some special properties:</p> <p>(i) they are closed under sum, </p> <p>(ii) they are closed under multiplication by positive scalars, </p> <p>(iii) spectrum of every matrix is positive, (all eigenvalues are nonnegative, and not all are equal to 0),</p> <p>(iv) $P+-P+iP+-iP=M_n(\mathbb{C})$.</p> <p>Does any other subset of matrix algebra $M_n(\mathbb{C})$ satisfy these properties except for $tPt^{-1}$, where $t$ is an invertible element in $M_n(\mathbb{C})$?</p> http://mathoverflow.net/questions/57719/algebraic-closure-of-commuting-pairs-of-matrices algebraic closure of commuting pairs of matrices spelas 2011-03-07T19:40:33Z 2011-03-07T21:05:51Z <p>Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M_n(F)×M_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.</p> <p>How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?</p> <p>Thank you.</p> http://mathoverflow.net/questions/120980/possible-restrictions-on-generators-of-m-n-mathbbc/120991#120991 Comment by 2013-02-06T18:32:23Z 2013-02-06T18:32:23Z Thanks. I have edited the question. http://mathoverflow.net/questions/120980/possible-restrictions-on-generators-of-m-n-mathbbc/120991#120991 Comment by 2013-02-06T17:23:06Z 2013-02-06T17:23:06Z Thank you for your example. But I would like to find $a$ and $b$ (or prove that they do not exist), such that $\alpha a+\beta b$ has the minimal polynomial of degree $&lt;n$ for all possible values $\alpha,\beta \in \mathbb{C}$. http://mathoverflow.net/questions/120980/possible-restrictions-on-generators-of-m-n-mathbbc/120989#120989 Comment by 2013-02-06T17:18:43Z 2013-02-06T17:18:43Z I know that there exist generators $a$ and $b$ where one of them has the minimal polynomial of degree $n$ (as also your example shows), but I am asking if it is always the case that at least for some $\alpha,\beta\in\mathbb{C}$ the minimal polynomial of $\alpha a+\beta b$ has degree $n$. http://mathoverflow.net/questions/116821/centralizer-in-a-matrix-algebra-over-commutative-polynomials Comment by 2013-02-06T15:46:28Z 2013-02-06T15:46:28Z Thank you. With this approach we are able to answer the question in affirmative. http://mathoverflow.net/questions/84863/centralizers-in-c-algebra/85243#85243 Comment by 2012-01-10T14:33:19Z 2012-01-10T14:33:19Z But it holds a+b=1, so b=f(a). http://mathoverflow.net/questions/84863/centralizers-in-c-algebra Comment by 2012-01-06T20:25:53Z 2012-01-06T20:25:53Z I am interested in the situation of general $C^*$-algebras, and some &quot;nontrivial&quot; correspondence between $a$ and $b$. Since the correspondence in $B(H)$ is quite &quot;strong&quot;, I expected that something nonobvious can be said also in general $C^*$-algebras. http://mathoverflow.net/questions/84863/centralizers-in-c-algebra Comment by 2012-01-05T09:26:50Z 2012-01-05T09:26:50Z Since the condition trivially holds if A is commutative, we exclude that case. In the noncommutative case, does it hold the same correspondence as in $B(H)$? http://mathoverflow.net/questions/84863/centralizers-in-c-algebra Comment by 2012-01-04T12:35:09Z 2012-01-04T12:35:09Z Yes, I am assuming that a and b are self-adjoint. Sorry for the mistake and thank you for pointing it out. http://mathoverflow.net/questions/79461/positive-hermitian-elements-in-m-n-mathbbc/79508#79508 Comment by 2011-10-30T16:02:50Z 2011-10-30T16:02:50Z I cannot find the question in Halmos books. There were some other useful information. Thank you. For case $n=2$, might Mathematica be able to compute this? http://mathoverflow.net/questions/79461/positive-hermitian-elements-in-m-n-mathbbc Comment by 2011-10-29T13:54:55Z 2011-10-29T13:54:55Z But $x^∗ax$ is also hermitian matrix if $a$ is. So $x^∗Px⊂P$, and $x^∗M_n(\mathbb{C})x=M_n(\mathbb{C})$ iff $x$ is invertible. So $x^∗Px$ either does not satisfy (iv) or equals $P$. http://mathoverflow.net/questions/57719/algebraic-closure-of-commuting-pairs-of-matrices/57730#57730 Comment by 2011-03-07T22:48:46Z 2011-03-07T22:48:46Z This helps a lot, thank you. http://mathoverflow.net/questions/57719/algebraic-closure-of-commuting-pairs-of-matrices Comment by 2011-03-07T20:18:53Z 2011-03-07T20:18:53Z @Victor: Sorry, I was reading this in an article where it was left as &quot;easy to check&quot;.