Syk Inhibition in MItigating Lung Allograft Rejection (SIMILAR): A Trial to Evaluate the Safety and Tolerability of Fostamatinib in Lung Transplant Patients With Donor-Specific Antibodies

* INCLUSION CRITERIA: Subjects who do not meet any of the following criteria during screening will not be randomized but will be counted toward study accrual. Screen failures may be rescreened at a later time if the reason for screening failure is revised. In order to be eligible to participate in this study, an individual must meet all of the following criteria: * First time LT recipients * Have provided signed written informed consent, prior to performing any study procedure, including screening procedures. * Age greater than or equal to 18 years * Patients who are DSA positive after day 21 post-transplantation. * Demonstrate no clinical or spirometry signs of allograft dysfunction at the time of randomization. * Have adequate liver function at the time of randomization, as defined by: * Serum aspartate aminotransferase (AST) \<=1.5 x Upper Limit of Normal (ULN) (unless the increased AST is assessed by the Investigator as due to hemolysis) and alanine aminotransferase (ALT) \<=1.5 x ULN. * Absolute neutrophil count \>=1.0 x 10\^9/L at the time of randomization. * Hemoglobin \>= 9 g/dL at the time of randomization. * For women of reproductive potential, have a negative serum pregnancy test during the screening period and at time of randomization. Women of reproductive potential are defined as sexually mature women who have not undergone a hysterectomy, bilateral oophorectomy, or tubal occlusion; or who have not been naturally postmenopausal (i.e., who have not menstruated at all for at least the preceding 1 year prior to signing informed consent unrelated to hormonal contraception). * For women of reproductive potential, be abstinent as part of their usual lifestyle, or agree to use 2 effective forms of contraception from the time of giving informed consent, during the study, and for 28 days following the last dose of study treatment. An effective form of contraception is defined as hormonal oral contraceptives, injectables, patches, intrauterine or subdermal contraceptive implants, and barrier methods. * Be willing to comply with all study procedures for the duration of the study. EXCLUSION CRITERIA: An individual who meets any of the following criteria will be excluded from participation in this study: * Have a significant medical condition that confers an unacceptable risk to participating in the study, and/or that could confound the interpretation of the study data. Such significant medical conditions include, but are not limited to the following: * Active CMV, EBV or other opportunistic infections * History of neutropenia (benign ethnic neutropenia and/or acquired neutropenia) within 90 days of screening. * History of posterior reversible encephalopathy syndrome (PRES) * History of poorly controlled hypertension or hypertensive crises (defined as systolic blood pressure \>=180 mmHg or average diastolic blood pressure \>=120 mmHg based on an average of 3 blood pressure readings despite adequate antihypertensive therapy) unless controlled for \>90 days at the time of randomization. * History of positive post-transplant active hepatitis C and/or hepatitis B viral infection. * History of drug-induced cholestatic hepatitis within 90 days of screening. * History of any primary malignancy within the last 5 years, with the exception of: curatively treated non-melanomatous skin cancer; curatively treated cervical or breast carcinoma in situ; or other primary tumor treated with curative intent, no known active disease present, and no treatment administered during the last 5 years. * History of positive human immunodeficiency virus 1 or 2 Ab with evidence of active infection, defined as (i.e., CD 4 count \<400/microliter and viral load \>100,000 copies/ml) while receiving antiretroviral therapy, based on available standard-of-care results. * Current or recent history of psychiatric disorder within the last 90 days that, in the opinion of the Investigator or Medical Monitor, could compromise the ability of the subject to cooperate with study visits and procedures. * Are currently enrolled in another therapeutic clinical trial involving ongoing therapy with any investigational or marketed product or placebo. * Having had a prior lung or any organ transplant. * Currently pregnant or lactating. * Estimated glomerular filtration (eGFR) rate less than 30 mL/min. * Any grade 3 diarrhea within 90 days at the time of randomization. * Subjects on strong CYP3A4 inducers. Glucocorticoids are standard transplant therapies and are not excluded. Relevant strong inducers include apalutamide, carbamazepine, encorafenib, enzalutamide, fosphenytoin, lumacaftor, lumacaftor-ivacaftor, mitotane, phenytoin, rifampin (rifampicin) - (https://www.uptodate.com/contents/image?imageKey=CARD%2F76992) -Subjects who have received prior treatment for DSA within 6 months of screening. Patients on an on-going therapy for first-time DSA are eligible, if screening is performed within 30 days of positive DSA results.