Notes on the Numerical Range
نویسنده
چکیده
This is an introduction to the notion of numerical range for bounded linear operators on Hilbert space. The main results are: determination of the numerical range for two by two matrices, the Toeplitz-Hausdorff Theorem establishing the convexity of the numerical range for any Hilbert space operator, and a detailed discussion of the relationship between the numerical range and the spectrum. The high points of this latter topic are: containment of the spectrum in the closure of the numerical range, and Hildebrandt’s theorem which asserts that the intersection of the closures of the numerical ranges of all operators similar to a given one T is the precisely the convex hull of the spectrum of T .
منابع مشابه
Evidence of the Potential Shift Mechanism in Crevice Corrosion (RESEARCH NOTES)
Previous research has indicated a qualitative resemblance between the current distribution along actively corroding crevice walls and currents measured in corresponding anodic polarization plots. Using detailed measurements of the potential gradient along the crevice, a semi-quantitative current distribution was calculated for the crevice wall using a double numerical differentiation method. Th...
متن کاملSome results on the block numerical range
The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.
متن کاملNotes on the genus Pileolaria (Pucciniales)
The results showed that Pileolaria terebinthi, type species of the genus Pileolaria is not a hemi–form or brachy–form rust species as mentioned among literature by several authors. Study on miscellaneous specimens collected from Iran and locus classicus of the species, France, revealed that P. terebinthi is an automacrocyclic rust and its aecial state is not uredinioi...
متن کاملPERRON-FROBENIUS THEORY ON THE NUMERICAL RANGE FOR SOME CLASSES OF REAL MATRICES
We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.
متن کاملOn the decomposable numerical range of operators
Let $V$ be an $n$-dimensional complex inner product space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $chi :Hrightarrow mathbb{C} $ is an irreducible character (not necessarily linear). Denote by $V_{chi}(H)$ the symmetry class of tensors associated with $H$ and $chi$. Let $K(T)in (V_{chi}(H))$ be the operator induced by $Tin text{End}(V)$. Th...
متن کاملC*-Algebra numerical range of quadratic elements
It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the C*-algebra numerical range.
متن کامل