Papers by Ramesh Garimella
Journal of the Indian Mathematical Society, Dec 1, 1999
Proceedings of Indian National Science Academy, 2015
ZARISKI TOPOLOGY AND SEPARATING IDEALS IN COMMUTATIVE BANACH ALGEBRAS
Proceedings of Indian National Science Academy, 2015
PRIME IDEALS AND THE IMAGE OF A DERIVATION ON A LOCAL BANACH ALGEBRA
For any given bounded linear operator A on a complex Hilbert space H, we give sufficient conditio... more For any given bounded linear operator A on a complex Hilbert space H, we give sufficient conditions to ensure the existence of a bounded operator B on H such that (i)AB + BA is of rank one, and (ii)I + e xP (A)+tQ(A) B is invertible for all x, t ∈ R where P (A) and Q(A) are polynomials in A. Our main results will provide a justiÞcation in general terms to a crucial step of the so-called operator method used by Aden, Carl, and Schiebold [1,3] to solve nonlinear partial differential equations like the Korteveg-deVries(KdV), modiÞed KdV, Kadomtsev-Petviashvili equations.
Bulletin of The Korean Mathematical Society, 1999
For a locally compact Abelian group G, and a commutative Banach algebra B, let (G, B) be the Bana... more For a locally compact Abelian group G, and a commutative Banach algebra B, let (G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is noncompact and B is a semiprime Banach algebras in which every minimal prime ideal is cnotained in a regular maximal ideal, then (G, B) contains no nontrivial separating idal. As a consequence we deduce some automatic continuity results for (G, B).
Proceedings of the American Mathematical Society, 2009
Let A be a bounded linear operator on a complex Banach space X. A problem, motivated by the opera... more Let A be a bounded linear operator on a complex Banach space X. A problem, motivated by the operator method used to solve integrable systems such as the Korteweg-deVries (KdV), modified KdV, sine-Gordon, and Kadomtsev-Petviashvili (KP) equations, is whether there exists a bounded linear operator B such that (i) AB + BA is of rank one, and (ii) (I + f (A)B) is invertible for every function f analytic in a neighborhood of the spectrum of A. We investigate solutions to this problem and discover an intriguing connection to the invariant subspace problem. Under the assumption that the convex hull of the spectrum of A does not contain 0, we show that there exists a solution B to (i) and (ii) if and only if A has a non-trivial invariant subspace.
ABSTRACT Suppose z1, z2, ... zn are complex numbers with absolute values more than 1 and Arg zj ?... more ABSTRACT Suppose z1, z2, ... zn are complex numbers with absolute values more than 1 and Arg zj ?Arg zk for j ? k where Arg w stands for the argument of the complex number w in [0,2p). In this note we show that $$\mathop {{\text{min}}}\limits_{\left| z \right| = 1} \frac{{\left| {\sum\limits_{j = 1}^n {\frac{{z_j }}{{z - z_j }}} } \right|}}{{\left| {\sum\limits_{j = 1}^n {\frac{1}{{z - z_j }}} } \right|}} \geqslant \frac{{\sum\limits_{j = 1}^n {\frac{{\left| {z_j } \right|}}{{\left| {z_j } \right| - 1}}} }}{{\sum\limits_{j - 1}^n {\frac{1}{{\left| {z_j } \right| - 1}}} }}.$$ We also give necessary and sufficient conditions for equality in the above inequality. As an application, we improve the result of Govil and Labelle on Bernstein's inequality for some special polynomials.
For any given square matrix A with complex entries, we give sufficient conditions to ensure the e... more For any given square matrix A with complex entries, we give sufficient conditions to ensure the existence of a square matrix B satisfying (i) AB + BA is of rank one, and (ii) I + e,(A)+tQ(A)B is invertible for all x, t 2 R where P(A) and Q(A) are polynomials in A.
International Journal of Mathematics and Mathematical Sciences, 2000
For a complete measure space(X,∑,μ), we give conditions which forceLp(X,μ), for1≤p<∞, to be is... more For a complete measure space(X,∑,μ), we give conditions which forceLp(X,μ), for1≤p<∞, to be isometrically isomorphic toℓp(Γ)for some index setΓwhich depends only on(X,μ). Also, we give some new characterizations which yield the inclusionLp(X,μ)⊂Lq(X,μ)for0<p<q.
Bulletin of the Australian Mathematical Society, 1997
For a locally compact Abelian group G and a commutative Banach algebra B, let L l (G,B) be the Ba... more For a locally compact Abelian group G and a commutative Banach algebra B, let L l (G,B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L X {G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L X (G,B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L l (G, B) and every epimorphism from a commutative Banach algebra onto L X (G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.
Abstract. For a complete measure space (X,Σ,µ), we give conditions which force Lp(X,µ), for 1 ≤ p... more Abstract. For a complete measure space (X,Σ,µ), we give conditions which force Lp(X,µ), for 1 ≤ p < ∞, to be isometrically isomorphic to p(Γ) for some index set Γ which depends only on (X,µ). Also, we give some new characterizations which yield the inclusion Lp(X,µ) ⊂ Lq(X,µ) for 0<p < q.
Linear Algebra and its Applications, 2006
For a given nonzero bounded linear operator A on a Banach space X, we show that if A or A * has a... more For a given nonzero bounded linear operator A on a Banach space X, we show that if A or A * has an eigenvalue then, except when the dimension of X is equal to two and the trace of A is zero, there exists a bounded linear operator B on X such that (i) AB + BA is of rank one, and (ii) I + f (A)B is invertible for every function f analytic in a neighborhood of the spectrum of A. This result was motivated by the operator method used by Carl et al. [H. Aden, B. Carl, On realizations of solutions of the KdV equation by determinants on operator ideals,
Proceedings of the American Mathematical Society, 1987
We characterize all closed subspaces of finite codimension in some specific types of function alg... more We characterize all closed subspaces of finite codimension in some specific types of function algebras e.g. these include C ( X ) C(X) : algebra of continous functions on a compact Hausdorff space, C n [ a , b ] {C^n}[a,b] : the algebra of n n -times continuously differentiable functions on the closed interval [ a , b ] [a,b] . Our work is a generalization of the well-known Gleason-Kahane-Želazko theorem [3, 6] for subspaces of codimension one in arbitrary unitary Banach algebras.
Mathematical Biosciences and Engineering, 2007
Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, ... more Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, the regulatory mechanism is analyzed mathematically. For this analysis, a mathematical model for the pathways is first established using a system of differential equations. Then the linear stability, controllability, and observability of the system are investigated. We show that the linearized system is controllable and observable, and that the real parts of all eigenvalues of the linearized system are nonpositive using Routh's stability criterion. Controllability and observability are structural properties of a dynamical system. Thus our results may explain why the metabolic pathways can be controlled and regulated. Finally observer-based and proportional output feedback controllers are designed to regulate the end product to its desired level. Applications to the regulation of blood glucose levels are discussed.
Proceedings of the American Mathematical Society, 1993
We prove that the separating ideal S ( D ) S(D) of any derivation D D on a commutative unital alg... more We prove that the separating ideal S ( D ) S(D) of any derivation D D on a commutative unital algebra B B is nilpotent if and only if S ( D ) ∩ ( ⋂ R n ) S(D) \cap (\bigcap {{R^n})} is a nil ideal, where R R is the Jacobson radical of B B . Also we show that any derivation D D on a commutative unital semiprime Banach algebra B B is continuous if and only if ⋂ ( S ( D ) ) n = { 0 } \bigcap {{{(S(D))}^n} = \{ 0\} } . Further we show that the set of all nilpotent elements of S ( D ) S(D) is equal to ⋂ ( S ( D ) ∩ P ) \bigcap {(S(D) \cap P)} , where the intersection runs over all nonclosed prime ideals of B B not containing S ( D ) S(D) . As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.
Proceedings of the American Mathematical Society, 1987
If every prime ideal is closed in a commutative semiprime Banach algebra with unit, then every de... more If every prime ideal is closed in a commutative semiprime Banach algebra with unit, then every derivation on it is continuous. Also if derivations are continuous on integral domains, then they are continuous on semiprime Banach algebras.
Missouri Journal of Mathematical Sciences
Missouri Journal of Mathematical Sciences
Missouri Journal of Mathematical Sciences
Uploads
Papers by Ramesh Garimella