The principal purpose of this announcement is to present an equivalent formulation of the invariant subspace conjecture for bounded linear operators acting on a Hubert space H. Specifically, the conjecture asserts that if B(H) denotes the algebra of bounded linear operators on H and AeB(H\ then A has a nontrivial invariant subspace. We show that the conjecture can be reduced to the study of operators having the property that their invariant subspaces are reducing spaces. In our earlier announcement of this result we called such an operator "completely normal" (cf.[2]); however, since then we have been convinced (by P. R. Halmos) that "reductive" is a more appropriate term. Our basic result is that if dim H > 1, then the invariant subspace conjecture is correct if, and only if, every reductive element of B{H) is normal. Inasmuch as the proof of the result requires an elaborate use of direct integral theory for rings of operators, we have not given proofs to the theorems. The complete proofs are expected to appear in a forthcoming monograph on direct integral theory and its applications. In particular we set H 0 = N(H) and let P 0 denote the projection of AMS 1970 subject classifications. Primary 47A15, 47C15; Secondary 46G10, 46J05.
Raman displacements, semiquantitative relative intensities, and quantitative depolarization factors are given for the above named hydrocarbons in the liquid state. The relative intensities and depolarization factors were obtained by use of a Gaertner microdensitometer. The depolarization factors were determined by a well-tested, single-exposure method. The previous data have been collected, tabulated, and compared with the present results, and probable values and average deviations are listed.
Purpose The aim of this study was to evaluate the relationship between the time from injury to ACL reconstruction (ACLR) and the rate as well as repairability of meniscal tears. Secondary aims were to evaluate the relationship between meniscal injury and Tegner Activity Scale, age, BMI, and gender. Methods Between 2012 and 2022, 1,840 consecutive ACLRs were performed. A total of 1,317 ACLRs were included with a mean patient age of 31.2 years ± 10.5 . Meniscal tear was assessed during arthroscopy using the ISAKOS classiication. Time from injury to ACLR, Tegner Activity Scale, age, BMI and gender were analysed in uni-and then in multivariate analyses. Patients were divided into four groups according to the time from injury to surgery: < 3 months (427; 32%), 3-6 months (388; 29%), 6-12 months (248; 19%) and > 12 months (254; 19%). Results Delaying ACLR > 12 months signiicantly increased the rate of medial meniscal (MM) injury (OR 1.14; p < 0.001). No correlation was found between a 3-or 6-month time from injury to surgery and MM tear. Performing ACLR > 3, 6, or 12 months after injury did not signiicantly increase the rate of lateral meniscal (LM) injury. Increasing Tegner activity scale was signiicantly associated with a lower rate of MM injury (OR 0.90; p = 0.020). An age > 30 years (OR 1.07; p = 0.025) and male gender (OR 1.13; p < 0.0001) was also associated with an increased rate of MM injury. Age > 30 years decreased the rate of MM repair (OR 0.85; p < 0.001). Male gender increased the rate of LM tear (OR 1.10; p = 0.001). Conclusion Performing ACLR more than 12 months after injury was associated with increased rates of MM injury but not with lower rates of repairable lesions. An increased pre-injury Tegner activity score was associated with a decreased rate of MM tear. Age > 30 years was associated with an increased rate of MM tear with concomitant ACL injury and a decreased rate of repairability of MM tear. ACLR should be performed within 12 months from injury to prevent from the risk of MM injury. Level of Evidence Level III.
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