2019 American College of Rheumatology/Arthritis Foundation Guideline for the Management of Osteoarthritis of the Hand, Hip, and Knee

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/153772/1/acr24131.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/153772/2/acr24131_am.pd

Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute. II

Let θ \theta denote the set of bounded linear operators T, acting on a separable Hilbert space K \mathcal {K} such that T ∗ T {T^\ast }T and T + T ∗ T + {T^\ast } commute. It is shown that such operators are G 1 {G_1} . A complete structure theory is developed for the case when σ ( T ) \sigma (T) does not intersect the real axis. Using this structure theory, several nonhyponormal operators in θ \theta with special properties are constructed.</p

On asymptotic properties of several classes of operators

Let p ( T , T ∗ ) p(T,{T^ \ast }) be a polynomial in T and T ∗ {T^ \ast } where T is a bounded linear operator on a separable Hilbert space. Let Δ = { T | p ( T , T ∗ ) = 0 } \Delta = \{ T|p(T,{T^ \ast }) = 0\} . Then Δ \Delta is said to be asymptotic for p if for every K &gt; 0 K &gt; 0 , there exists an ε 0 &gt; 0 {\varepsilon _0} &gt; 0 and function δ ( ε , K ) , lim ε → 0 δ ( ε , K ) = 0 \delta (\varepsilon ,K),{\lim _{\varepsilon \to 0}}\delta (\varepsilon ,K) = 0 , such that if ε &gt; ε 0 , ‖ T ‖ ⩽ K \varepsilon &gt; {\varepsilon _0},\left \| T \right \| \leqslant K , and ‖ p ( T , T ∗ ) ‖ &gt; ε \left \| {p(T,{T^ \ast })} \right \| &gt; \varepsilon , then there exists T ^ ∈ Δ \hat T \in \Delta such that ‖ T − T ^ ‖ &gt; δ ( ε , K ) \left \| {T - \hat T} \right \| &gt; \delta (\varepsilon ,K) . It is observed that the hermitian, unitary, and isometric operators are asymptotic for the obvious polynomials. It is known that the normals are not asymptotic for p ( T , T ∗ ) = T ∗ T − T T ∗ p(T,{T^ \ast }) = {T^ \ast }T - T{T^ \ast } . An example gives several negative results including one that says the quasinormals are not asymptotic for p ( T , T ∗ ) = T T ∗ T − T ∗ T 2 p(T,{T^\ast }) = T{T^\ast }T - {T^\ast }{T^2} . It is shown that if p is any polynomial in just one of T or T ∗ {T^ \ast } , then Δ \Delta is asymptotic for p.</p